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"type": "text",
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"text": "2026 ICM Problem D: Managing Sports for Success",
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"img_path": "images/0af13f43c25137d8e22b7c835aa3d8dc127f8498eea06196deb71b9e795dbf0c.jpg",
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"text": "\"The player's job is to help his team win.\" - Cliff Blau, baseball historian and statistician \"The player's job is to make money for the owner.\" - all sports team owners",
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"text": "Fans tend to focus on the players on the field or court, but that is only the tip of the sports business iceberg. Sports are entertainment first and foremost. Entertainment is a profit-generating business and the players are hired mostly for that purpose. Often, fans of spectator sports ignore the financial purpose of a sport and try to focus on the game itself and its participants. However, in professional sports business, the primary goal is to make money for the owner and not necessarily win games. While these two goals may be related, since winning generates more interest in the team, other factors are involved. And for some sports teams, there are crucial moments when opportunity and risk are both high – like this year's situation for teams in the Women's National Basketball Association (WNBA), the most prominent women's professional basketball league in the United States. For many reasons (especially higher fan interest), WNBA teams are hoping to evolve from risky startup businesses into major entertainment enterprises by taking advantage of increased media attention, new team franchises, larger venues, and a new digital platform to increase revenue. The owners in that league need to use sports analytics to succeed on the court but also use financial modeling to achieve significant financial gain in the bottom line of their business's profit sheet.",
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"text": "Should players (and other employees of the team business) in a sport get paid more for their performance that produces wins or for their contributions in turning a profit for the team owner? Sometimes, a player's sport performance is directly related to profit, but not always. Some players may attract fans based more on popularity than performance. These players may generate ticket, parking, concession, and jersey revenue much more than players with higher levels of performance. Financial and sports analytics models need to connect to create good team decision making.",
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"text": "In the emerging field of sports analytics with various kinds and amounts of performance data, there continue to be challenges to build statistics that quantify the value of player talents and performances (what statistic to measure, how to measure it, when to measure it). Some players are injured more frequently than others. How does that affect player value? Some have personalities that lead to more popularity and appeal that lead to financial gain. Context and timing matter in the sense that some players, even those with average performance, come through at important moments of the game or critical times in the season. There is a temporal element that must consider the measure of future potential of a player/employee on achieving the goal of the team. Some roles may be performance or skill-based and other roles are accomplished more by hard work and perseverance.",
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"type": "footer",
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"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org",
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"text": "The player or team perceptions, popularity, timing, and marketing can play major roles, in addition to the location of the team. Teams in large markets often have different sports situations and goals than small-market teams. Those differences impact how owners achieve profit and recruit their players and employees. Can modeling help an owner establish methodologies for setting offers, negotiating, and writing contracts?",
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"text": "There are many team issues that are strictly or mostly financial, just as there are issues that are mostly sports. In many cases, professional sports teams are franchises that are part of the league enterprise and often operate with additional rules and constraints set by leagues or governments on their player salaries and contracts. These are intended to make the game competition fair with some reasonable amount of competitive balance. Some professional sports have systems that regulate salaries with caps or taxes. Every season, the owner must decide how much to finance with debt versus equity and whether risks in the form of seeking better team performance with associated additional costs are worth taking. In the sports business world, conditions such as revenues, salaries, injuries, trade opportunities, taxes, fees, and interest rates change over time. Sports teams are now seen as premium assets, with values in many sports soaring far beyond historical norms due to financial and market factors such as lucrative media deals and accumulation of vast data streams and intellectual property.",
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"type": "text",
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"text": "As a modeling group for a sports team, your ICM team can use publicly available sport and finance data for a team of your choice (the team you select must consist of at least 5 players that play cooperatively at the same time and be a member of a professional league) and build a business and management model for the team for the coming or next season.",
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"text": "As was mentioned earlier as an example of how this modeling work can be extremely valuable, the WNBA is undergoing significant financial changes -- record viewership, rising franchise values, and significant player benefit expectations. Currently, negotiations and demands over the revenue-sharing agreement between teams and players are sticking points. During this coming season, team owners have an opportunity to remake and improve their business or succumb to risks that may cause them to sell or take on substantial debt. These issues create a situation where solid financial and sport modeling can make a big difference for the current and future owners of these teams. You may use a WNBA team if you care to, but you are not required to do that.",
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"type": "text",
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"text": "Questions to consider:",
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"type": "text",
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"text": "Design a dynamic decision-making model that would help your team owner and general managers adjust their leverage in response to changing team performance and economic conditions. The goal is to maximize team profit and value while managing team structure and performance. The model should include priorities and actions for the management teams in both business operations and team operations, and account for systems that will help the owner make decisions through the coming season and beyond.",
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"text": "Based on the needs of the team and your model, develop a strategy to acquire players for next season using the standard practice for your team's league such as a draft, free agency, trades, transfer fees, or other standard practices. You may want to consider how to value a player or the",
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"type": "footer",
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"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org",
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"text": "team dynamics in terms of the profit for the team owners. Using the outcomes of your model, discuss the strengths and weaknesses of your strategy on the business.",
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"text": "There are many league-determined rulings that impact how a single team can operate, such as salary caps, number of players on a roster, schedule (number, order, location, and date of games in a season, so consequently days of rest), media contracts and rights, revenue distributions, and others. If a league is expanding the number of franchises (such as for WNBA), it is likely to impact all teams in the league. Use your model to decide how your team's strategy should change from your initial strategy during a season with league expansion. How does the location for the new team impact your model and resulting strategy? Be clear on the impact on the team owners and locations for the new team that would be particularly harmful or beneficial under an expansion.",
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"type": "text",
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"text": "Consider one additional business decision and use your model to design the best strategy for your team. Some examples include but are not limited to:",
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"text": "- Ticket sales vary greatly by the size of the stadium, time of year, popularity of the team (yours and opponent), size of the team's market, and other factors. A team may choose to maximize ticket sale revenue for each game or lower the prices to have larger attendance with the possibility to convert some of those attendees into season ticket holders. How do you determine the optimal ticket pricing strategy over a season?",
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"text": "- The venue for the team to play its games may be rented or owned with the need to maintain, renovate, or even build a new venue. How do you balance the long-term cost of the venue when it is a short-term decision?",
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"type": "text",
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"text": "- Player equity in ownership can be one strategy for subsidizing large salaries, such as revenue sharing (single season), profit participation (bonus), decision makers (as part of unions or collective bargaining), long-term equity stake (part owner), or other methods. Player equity options need to be sufficiently lucrative for a player to accept it, but not undermine the future funding options. How do you determine which players, if any, are offered equity and how much?",
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"type": "text",
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"text": "- Media deals are a large source of revenue, fan engagement, and brand building, often producing high engagement and advertising potential. While leagues usually contract national deals, teams can sometimes broker their own local deals or streaming options. Does your team need to improve or change its media presence?",
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"bbox": [
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"type": "text",
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"text": "- Division or conference structure, which can build or take advantage of rivalries where rival teams play more often, is generally determined by the league. Are there ways that league structures and schedules be reconfigured to increase profit for your team?",
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"bbox": [
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},
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{
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"type": "text",
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"text": "- Determine your own issue that applies to your team or sport and use your model to help decide the issue to improve team performance or owner profit.",
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},
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{
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"type": "footer",
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"text": "©2026 by COMAP | www.mathmodels.org | www.mathmodels.org | info@comap.org",
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"bbox": [
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{
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"type": "text",
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"text": "How does your model help management adjust when a key player is injured?",
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},
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{
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"type": "text",
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"text": "Write a one- to two-page letter to your team's owner and general manager that summarizes your recommended strategy, discusses trade-offs and risks, and reflects on how your plan supports both competitive success and financial health.",
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"bbox": [
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},
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{
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"type": "text",
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"text": "Your PDF solution of no more than 25 total pages should include:",
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"bbox": [
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"page_idx": 3
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},
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{
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"type": "list",
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"sub_type": "text",
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"list_items": [
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"One-page Summary Sheet.",
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"Table of Contents.",
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"- Your complete solution.",
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"One-to-Two-Page Letter.",
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"- References List.",
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"AI Use Report (If used does not count toward the 25-page limit.)"
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],
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"bbox": [
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{
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"type": "text",
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"text": "Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.",
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"bbox": [
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},
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{
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"type": "text",
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"text": "Glossary",
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"text_level": 1,
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"bbox": [
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"page_idx": 3
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},
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{
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"type": "text",
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"text": "Competitive balance is how evenly matched the teams are in a league or competition.",
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{
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"type": "text",
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"text": "Draft is a way for a sports league to assign new players to teams in an organized manner.",
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"content": "\"The player's job is to help his team win.\" - Cliff Blau, baseball historian and statistician \"The player's job is to make money for the owner.\" - all sports team owners"
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"content": "Should players (and other employees of the team business) in a sport get paid more for their performance that produces wins or for their contributions in turning a profit for the team owner? Sometimes, a player's sport performance is directly related to profit, but not always. Some players may attract fans based more on popularity than performance. These players may generate ticket, parking, concession, and jersey revenue much more than players with higher levels of performance. Financial and sports analytics models need to connect to create good team decision making."
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"content": "The player or team perceptions, popularity, timing, and marketing can play major roles, in addition to the location of the team. Teams in large markets often have different sports situations and goals than small-market teams. Those differences impact how owners achieve profit and recruit their players and employees. Can modeling help an owner establish methodologies for setting offers, negotiating, and writing contracts?"
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"content": "There are many team issues that are strictly or mostly financial, just as there are issues that are mostly sports. In many cases, professional sports teams are franchises that are part of the league enterprise and often operate with additional rules and constraints set by leagues or governments on their player salaries and contracts. These are intended to make the game competition fair with some reasonable amount of competitive balance. Some professional sports have systems that regulate salaries with caps or taxes. Every season, the owner must decide how much to finance with debt versus equity and whether risks in the form of seeking better team performance with associated additional costs are worth taking. In the sports business world, conditions such as revenues, salaries, injuries, trade opportunities, taxes, fees, and interest rates change over time. Sports teams are now seen as premium assets, with values in many sports soaring far beyond historical norms due to financial and market factors such as lucrative media deals and accumulation of vast data streams and intellectual property."
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"content": "As a modeling group for a sports team, your ICM team can use publicly available sport and finance data for a team of your choice (the team you select must consist of at least 5 players that play cooperatively at the same time and be a member of a professional league) and build a business and management model for the team for the coming or next season."
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"content": "As was mentioned earlier as an example of how this modeling work can be extremely valuable, the WNBA is undergoing significant financial changes -- record viewership, rising franchise values, and significant player benefit expectations. Currently, negotiations and demands over the revenue-sharing agreement between teams and players are sticking points. During this coming season, team owners have an opportunity to remake and improve their business or succumb to risks that may cause them to sell or take on substantial debt. These issues create a situation where solid financial and sport modeling can make a big difference for the current and future owners of these teams. You may use a WNBA team if you care to, but you are not required to do that."
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"angle": 0,
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"content": "Questions to consider:"
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},
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"angle": 0,
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"content": "Design a dynamic decision-making model that would help your team owner and general managers adjust their leverage in response to changing team performance and economic conditions. The goal is to maximize team profit and value while managing team structure and performance. The model should include priorities and actions for the management teams in both business operations and team operations, and account for systems that will help the owner make decisions through the coming season and beyond."
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"content": "Based on the needs of the team and your model, develop a strategy to acquire players for next season using the standard practice for your team's league such as a draft, free agency, trades, transfer fees, or other standard practices. You may want to consider how to value a player or the"
|
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|
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"angle": 0,
|
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|
"content": "team dynamics in terms of the profit for the team owners. Using the outcomes of your model, discuss the strengths and weaknesses of your strategy on the business."
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
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"angle": 0,
|
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|
"content": "There are many league-determined rulings that impact how a single team can operate, such as salary caps, number of players on a roster, schedule (number, order, location, and date of games in a season, so consequently days of rest), media contracts and rights, revenue distributions, and others. If a league is expanding the number of franchises (such as for WNBA), it is likely to impact all teams in the league. Use your model to decide how your team's strategy should change from your initial strategy during a season with league expansion. How does the location for the new team impact your model and resulting strategy? Be clear on the impact on the team owners and locations for the new team that would be particularly harmful or beneficial under an expansion."
|
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},
|
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],
|
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|
"angle": 0,
|
||||||
|
"content": "Consider one additional business decision and use your model to design the best strategy for your team. Some examples include but are not limited to:"
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
|
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"bbox": [
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],
|
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|
"angle": 0,
|
||||||
|
"content": "- Ticket sales vary greatly by the size of the stadium, time of year, popularity of the team (yours and opponent), size of the team's market, and other factors. A team may choose to maximize ticket sale revenue for each game or lower the prices to have larger attendance with the possibility to convert some of those attendees into season ticket holders. How do you determine the optimal ticket pricing strategy over a season?"
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- The venue for the team to play its games may be rented or owned with the need to maintain, renovate, or even build a new venue. How do you balance the long-term cost of the venue when it is a short-term decision?"
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Player equity in ownership can be one strategy for subsidizing large salaries, such as revenue sharing (single season), profit participation (bonus), decision makers (as part of unions or collective bargaining), long-term equity stake (part owner), or other methods. Player equity options need to be sufficiently lucrative for a player to accept it, but not undermine the future funding options. How do you determine which players, if any, are offered equity and how much?"
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
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],
|
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|
"angle": 0,
|
||||||
|
"content": "- Media deals are a large source of revenue, fan engagement, and brand building, often producing high engagement and advertising potential. While leagues usually contract national deals, teams can sometimes broker their own local deals or streaming options. Does your team need to improve or change its media presence?"
|
||||||
|
},
|
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|
{
|
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|
"type": "text",
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|
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"angle": 0,
|
||||||
|
"content": "- Division or conference structure, which can build or take advantage of rivalries where rival teams play more often, is generally determined by the league. Are there ways that league structures and schedules be reconfigured to increase profit for your team?"
|
||||||
|
},
|
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|
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|
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"angle": 0,
|
||||||
|
"content": "- Determine your own issue that applies to your team or sport and use your model to help decide the issue to improve team performance or owner profit."
|
||||||
|
},
|
||||||
|
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|
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|
"type": "footer",
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"content": "©2026 by COMAP | www.mathmodels.org | www.mathmodels.org | info@comap.org"
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|
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"angle": 0,
|
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|
"content": "How does your model help management adjust when a key player is injured?"
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
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],
|
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|
"angle": 0,
|
||||||
|
"content": "Write a one- to two-page letter to your team's owner and general manager that summarizes your recommended strategy, discusses trade-offs and risks, and reflects on how your plan supports both competitive success and financial health."
|
||||||
|
},
|
||||||
|
{
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],
|
||||||
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"angle": 0,
|
||||||
|
"content": "Your PDF solution of no more than 25 total pages should include:"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
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"bbox": [
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],
|
||||||
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"angle": 0,
|
||||||
|
"content": "One-page Summary Sheet."
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
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"bbox": [
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"angle": 0,
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"content": "Table of Contents."
|
||||||
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},
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||||||
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"type": "text",
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"angle": 0,
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"content": "- Your complete solution."
|
||||||
|
},
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{
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"type": "text",
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"content": "- References List."
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"content": "Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution."
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"angle": 0,
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"content": "Glossary"
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"content": "Competitive balance is how evenly matched the teams are in a league or competition."
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"angle": 0,
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"content": "Draft is a way for a sports league to assign new players to teams in an organized manner."
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"content": "Free agency is a system that allows players to choose which team they will play for after their contract with a team expires."
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"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org |"
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|
|||||||
|
# 2026 ICM Problem D: Managing Sports for Success
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
"The player's job is to help his team win." - Cliff Blau, baseball historian and statistician "The player's job is to make money for the owner." - all sports team owners
|
||||||
|
|
||||||
|
Fans tend to focus on the players on the field or court, but that is only the tip of the sports business iceberg. Sports are entertainment first and foremost. Entertainment is a profit-generating business and the players are hired mostly for that purpose. Often, fans of spectator sports ignore the financial purpose of a sport and try to focus on the game itself and its participants. However, in professional sports business, the primary goal is to make money for the owner and not necessarily win games. While these two goals may be related, since winning generates more interest in the team, other factors are involved. And for some sports teams, there are crucial moments when opportunity and risk are both high – like this year's situation for teams in the Women's National Basketball Association (WNBA), the most prominent women's professional basketball league in the United States. For many reasons (especially higher fan interest), WNBA teams are hoping to evolve from risky startup businesses into major entertainment enterprises by taking advantage of increased media attention, new team franchises, larger venues, and a new digital platform to increase revenue. The owners in that league need to use sports analytics to succeed on the court but also use financial modeling to achieve significant financial gain in the bottom line of their business's profit sheet.
|
||||||
|
|
||||||
|
Should players (and other employees of the team business) in a sport get paid more for their performance that produces wins or for their contributions in turning a profit for the team owner? Sometimes, a player's sport performance is directly related to profit, but not always. Some players may attract fans based more on popularity than performance. These players may generate ticket, parking, concession, and jersey revenue much more than players with higher levels of performance. Financial and sports analytics models need to connect to create good team decision making.
|
||||||
|
|
||||||
|
In the emerging field of sports analytics with various kinds and amounts of performance data, there continue to be challenges to build statistics that quantify the value of player talents and performances (what statistic to measure, how to measure it, when to measure it). Some players are injured more frequently than others. How does that affect player value? Some have personalities that lead to more popularity and appeal that lead to financial gain. Context and timing matter in the sense that some players, even those with average performance, come through at important moments of the game or critical times in the season. There is a temporal element that must consider the measure of future potential of a player/employee on achieving the goal of the team. Some roles may be performance or skill-based and other roles are accomplished more by hard work and perseverance.
|
||||||
|
|
||||||
|
The player or team perceptions, popularity, timing, and marketing can play major roles, in addition to the location of the team. Teams in large markets often have different sports situations and goals than small-market teams. Those differences impact how owners achieve profit and recruit their players and employees. Can modeling help an owner establish methodologies for setting offers, negotiating, and writing contracts?
|
||||||
|
|
||||||
|
There are many team issues that are strictly or mostly financial, just as there are issues that are mostly sports. In many cases, professional sports teams are franchises that are part of the league enterprise and often operate with additional rules and constraints set by leagues or governments on their player salaries and contracts. These are intended to make the game competition fair with some reasonable amount of competitive balance. Some professional sports have systems that regulate salaries with caps or taxes. Every season, the owner must decide how much to finance with debt versus equity and whether risks in the form of seeking better team performance with associated additional costs are worth taking. In the sports business world, conditions such as revenues, salaries, injuries, trade opportunities, taxes, fees, and interest rates change over time. Sports teams are now seen as premium assets, with values in many sports soaring far beyond historical norms due to financial and market factors such as lucrative media deals and accumulation of vast data streams and intellectual property.
|
||||||
|
|
||||||
|
As a modeling group for a sports team, your ICM team can use publicly available sport and finance data for a team of your choice (the team you select must consist of at least 5 players that play cooperatively at the same time and be a member of a professional league) and build a business and management model for the team for the coming or next season.
|
||||||
|
|
||||||
|
As was mentioned earlier as an example of how this modeling work can be extremely valuable, the WNBA is undergoing significant financial changes -- record viewership, rising franchise values, and significant player benefit expectations. Currently, negotiations and demands over the revenue-sharing agreement between teams and players are sticking points. During this coming season, team owners have an opportunity to remake and improve their business or succumb to risks that may cause them to sell or take on substantial debt. These issues create a situation where solid financial and sport modeling can make a big difference for the current and future owners of these teams. You may use a WNBA team if you care to, but you are not required to do that.
|
||||||
|
|
||||||
|
# Questions to consider:
|
||||||
|
|
||||||
|
Design a dynamic decision-making model that would help your team owner and general managers adjust their leverage in response to changing team performance and economic conditions. The goal is to maximize team profit and value while managing team structure and performance. The model should include priorities and actions for the management teams in both business operations and team operations, and account for systems that will help the owner make decisions through the coming season and beyond.
|
||||||
|
|
||||||
|
Based on the needs of the team and your model, develop a strategy to acquire players for next season using the standard practice for your team's league such as a draft, free agency, trades, transfer fees, or other standard practices. You may want to consider how to value a player or the
|
||||||
|
|
||||||
|
team dynamics in terms of the profit for the team owners. Using the outcomes of your model, discuss the strengths and weaknesses of your strategy on the business.
|
||||||
|
|
||||||
|
There are many league-determined rulings that impact how a single team can operate, such as salary caps, number of players on a roster, schedule (number, order, location, and date of games in a season, so consequently days of rest), media contracts and rights, revenue distributions, and others. If a league is expanding the number of franchises (such as for WNBA), it is likely to impact all teams in the league. Use your model to decide how your team's strategy should change from your initial strategy during a season with league expansion. How does the location for the new team impact your model and resulting strategy? Be clear on the impact on the team owners and locations for the new team that would be particularly harmful or beneficial under an expansion.
|
||||||
|
|
||||||
|
Consider one additional business decision and use your model to design the best strategy for your team. Some examples include but are not limited to:
|
||||||
|
|
||||||
|
- Ticket sales vary greatly by the size of the stadium, time of year, popularity of the team (yours and opponent), size of the team's market, and other factors. A team may choose to maximize ticket sale revenue for each game or lower the prices to have larger attendance with the possibility to convert some of those attendees into season ticket holders. How do you determine the optimal ticket pricing strategy over a season?
|
||||||
|
|
||||||
|
- The venue for the team to play its games may be rented or owned with the need to maintain, renovate, or even build a new venue. How do you balance the long-term cost of the venue when it is a short-term decision?
|
||||||
|
|
||||||
|
- Player equity in ownership can be one strategy for subsidizing large salaries, such as revenue sharing (single season), profit participation (bonus), decision makers (as part of unions or collective bargaining), long-term equity stake (part owner), or other methods. Player equity options need to be sufficiently lucrative for a player to accept it, but not undermine the future funding options. How do you determine which players, if any, are offered equity and how much?
|
||||||
|
|
||||||
|
- Media deals are a large source of revenue, fan engagement, and brand building, often producing high engagement and advertising potential. While leagues usually contract national deals, teams can sometimes broker their own local deals or streaming options. Does your team need to improve or change its media presence?
|
||||||
|
|
||||||
|
- Division or conference structure, which can build or take advantage of rivalries where rival teams play more often, is generally determined by the league. Are there ways that league structures and schedules be reconfigured to increase profit for your team?
|
||||||
|
|
||||||
|
- Determine your own issue that applies to your team or sport and use your model to help decide the issue to improve team performance or owner profit.
|
||||||
|
|
||||||
|
How does your model help management adjust when a key player is injured?
|
||||||
|
|
||||||
|
Write a one- to two-page letter to your team's owner and general manager that summarizes your recommended strategy, discusses trade-offs and risks, and reflects on how your plan supports both competitive success and financial health.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One-to-Two-Page Letter.
|
||||||
|
- References List.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
Competitive balance is how evenly matched the teams are in a league or competition.
|
||||||
|
|
||||||
|
Draft is a way for a sports league to assign new players to teams in an organized manner.
|
||||||
|
|
||||||
|
Free agency is a system that allows players to choose which team they will play for after their contract with a team expires.
|
||||||
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"type": "text",
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"text": "2026 ICM Problem E: Passive Solar Shading",
|
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"text": "Passive solar shading has become a common addition to both housing and commercial buildings as a part of a retrofit or in new construction. It is relatively inexpensive and creates cost savings in heating and cooling. The shades are designed to block summer sun from entering a building, while allowing winter sun to not only enter the building but to warm a thermal mass that can reradiate for many hours after. Strategies such as overhangs, vegetative shading, brise-soleil systems, and high-performance glazing can reduce heat gain in buildings during higher temperatures.",
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"text": "Passive solar shading is different depending on building orientation, window area distribution between the different faces of the building, and climate. It also requires the presence of an internal thermal mass that can be heated by the direct sun. This thermal mass can be concrete, stone, water, or other material that can store heat. The thermal mass not only stores heat but reduces temperature swings throughout the day.",
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"text": "These techniques use the predictable path of the sun (determined through the use of solar position calculators), materials, geometry, and natural environmental conditions to maintain comfort and reduce energy consumption. However, the typical calculations make use of the angle of the sun at solar noon on the Summer and Winter Solstices to calculate the optimal extension of a shade over a window as shown in Figure 1. This is a simplistic view of the problem, and future metrics must do better to account for change.",
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|
||||||
|
"image_caption": [
|
||||||
|
"Figure 1: Passive Solar Shading - Winter and Summer Sun on Solstices"
|
||||||
|
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|
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|
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"text": "You have been hired by the Collective Organizations Making Astrophysical Protections (COMAP) to innovate the next generation of solar shading strategies to be implemented at the notional Sungrove University and notional Borealis University.",
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{
|
||||||
|
"type": "text",
|
||||||
|
"text": "The notional Sungrove University, located in a warm, low-latitude region with high solar exposure and increasingly frequent heat waves, is planning a major transformation of its main academic quad. The campus currently suffers from excessive cooling costs and glare in the classrooms. The university leadership has decided to pursue a net-zero cooling initiative by 2040.",
|
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"bbox": [
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{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Notably, Sungrove University is planning to retrofit its Academic Hall North. It is a two-story classroom and office building. The interior layout combines perimeter offices and classrooms with interior corridors. The building has a rectangular footprint (60m × 24m) with its long side aligned east-west as shown in Figure 2. The facade consists of double glazing and a brick veneer with an average window-to-wall ratio of $45\\%$ on the south facing side and $30\\%$ on the remaining sides. The building relies on mechanical cooling in the summer and hydronic heating in the winter with limited passive strategies in place. Additional features of this notional building are yours to imagine. Ensure you communicate these features in your writing to COMAP.",
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||||||
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"image_caption": [
|
||||||
|
"Figure 2: Academic Hall North footprint"
|
||||||
|
],
|
||||||
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|
||||||
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"type": "text",
|
||||||
|
"text": "Additionally, COMAP has been hired by the notional Borealis University, located at a high latitude where winter temperatures remain below freezing for months, sunlight hours are limited, and buildings experience heavy heating demands.",
|
||||||
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||||||
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"text": "Sungrove University and Borealis University are also both planning a new student union that will serve as the hub of university activities. They have each mandated that their new student union building relies heavily on passive solar shading rather than mechanical cooling systems. The Universities want their student union building to serve as a prototype for future developments, meaning that their passive solar strategy design must perform well not only today, but under projected climate conditions well into the future.",
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|
||||||
|
"type": "text",
|
||||||
|
"text": "Beyond the standard approach to shading as outlined in the Background, to assist these notional universities, you should extend your ideas to include:",
|
||||||
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||||||
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|
||||||
|
"list_items": [
|
||||||
|
"- Shading needs throughout the day rather than just at solar noon.",
|
||||||
|
"- Windows of different sizes and shapes.",
|
||||||
|
"- Windows that do not face exactly south/north (depending on the hemisphere).",
|
||||||
|
"- Shades of different styles and materials that would match the architecture of the building."
|
||||||
|
],
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||||||
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"text": "©2026 by COMAP | www.mathmodels.org | info@comap.org |",
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|
||||||
|
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|
||||||
|
"text": "As with any new strategy or model, you will not only need to describe your approach but also explain the advantages that your proposal holds over the previous standard. COMAP needs to know how your passive solar shading strategies can more effectively reduce heat gain in campus buildings during the summer while still admitting beneficial winter sun.",
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||||||
|
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|
||||||
|
"type": "text",
|
||||||
|
"text": "Requirements",
|
||||||
|
"text_level": 1,
|
||||||
|
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|
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|
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|
||||||
|
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|
||||||
|
"text": "Your team has been asked by COMAP to provide a model-based feasibility analysis that determines how Sungrove University can reduce its academic year cooling load with passive solar design in the retrofit of buildings on campus. To do so, design a retrofit for Sungrove University's Academic Hall North that optimizes heating and cooling throughout the academic year. What passive solar strategies and building features would you use, and how would you evaluate their performance?",
|
||||||
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||||||
|
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|
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|
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|
||||||
|
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|
||||||
|
"text": "Borealis University has a building with a similar design to Sungrove University's Academic Hall North. How can extending your work for Sungrove University to include the crucial importance of the effective use of a thermal mass provide Borealis University with a plan to use passive solar shading? You may want to consider building geometry, material selection, glazing positioning, internal thermal mass, or other aspects to maximize winter heat gain while avoiding overheating in the warmer months.",
|
||||||
|
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|
||||||
|
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|
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|
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|
||||||
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||||||
|
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|
||||||
|
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|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "The retrofit design models at both Sungrove and Borealis Universities are helpful for only those notional sites. Adapt your model and discuss the design considerations for other locations including the different heating and cooling needs at places that might have similar latitudes.",
|
||||||
|
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|
||||||
|
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|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Design a passive solar shading strategy for the new student union building at either Sungrove University or Borealis University that keeps the building temperate. Describe the strategies, building features, and modeling approaches you would use to evaluate performance over time. You may wish to address some of the following in your analysis:",
|
||||||
|
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|
||||||
|
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|
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|
||||||
|
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|
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|
||||||
|
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|
||||||
|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
"sub_type": "text",
|
||||||
|
"list_items": [
|
||||||
|
"- Predicting solar heat gain",
|
||||||
|
"- Estimating heating and/or cooling load reductions",
|
||||||
|
"- Accounting for seasonal variations",
|
||||||
|
"- Evaluating the tradeoffs between daylighting needs and shading effectiveness"
|
||||||
|
],
|
||||||
|
"bbox": [
|
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||||||
|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Write a one-to-two-page letter to either Sungrove University or Borealis University (not both) outlining the steps they should take to include passive solar shading in both their retrofit and new building plans.",
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
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|
||||||
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|
||||||
|
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|
||||||
|
],
|
||||||
|
"page_idx": 2
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Your PDF solution of no more than 25 total pages should include:",
|
||||||
|
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|
||||||
|
112,
|
||||||
|
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|
||||||
|
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|
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|
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|
||||||
|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
"sub_type": "text",
|
||||||
|
"list_items": [
|
||||||
|
"One-page Summary Sheet.",
|
||||||
|
"Table of Contents.",
|
||||||
|
"- Your complete solution.",
|
||||||
|
"One-to-Two-Page Letter.",
|
||||||
|
"- References List.",
|
||||||
|
"AI Use Report (If used does not count toward the 25-page limit.)"
|
||||||
|
],
|
||||||
|
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|
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|
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|
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|
||||||
|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "footer",
|
||||||
|
"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org",
|
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|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.",
|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Glossary",
|
||||||
|
"text_level": 1,
|
||||||
|
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|
||||||
|
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|
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|
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|
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|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Solar noon is the moment during the day when the Sun is at its highest point in the sky for a given location.",
|
||||||
|
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|
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|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Winter Solstice is the day with the least daylight of the year, caused by Earth's tilt.",
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
|
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|
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|
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|
||||||
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|
||||||
|
"page_idx": 3
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Summer Solstice is the day with the most daylight of the year, caused by Earth's tilt.",
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
|
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|
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|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Notional means theoretical or fictitious. The universities in this problem are not real, but only theoretical case studies.",
|
||||||
|
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|
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|
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|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Net-zero cooling means providing cooling without adding greenhouse gases to the atmosphere.",
|
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|
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|
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|
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|
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|
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|
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|
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|
||||||
|
{
|
||||||
|
"type": "footer",
|
||||||
|
"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org",
|
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|
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|
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|
||||||
|
]
|
||||||
@@ -0,0 +1,593 @@
|
|||||||
|
[
|
||||||
|
[
|
||||||
|
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|
||||||
|
"type": "title",
|
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|
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|
"content": "2026 ICM Problem E: Passive Solar Shading"
|
||||||
|
},
|
||||||
|
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|
||||||
|
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||||||
|
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|
||||||
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|
||||||
|
"content": null
|
||||||
|
},
|
||||||
|
{
|
||||||
|
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|
||||||
|
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||||||
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|
||||||
|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
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|
||||||
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||||||
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||||||
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||||||
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|
||||||
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|
||||||
|
"angle": 0,
|
||||||
|
"content": "Passive solar shading has become a common addition to both housing and commercial buildings as a part of a retrofit or in new construction. It is relatively inexpensive and creates cost savings in heating and cooling. The shades are designed to block summer sun from entering a building, while allowing winter sun to not only enter the building but to warm a thermal mass that can reradiate for many hours after. Strategies such as overhangs, vegetative shading, brise-soleil systems, and high-performance glazing can reduce heat gain in buildings during higher temperatures."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
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|
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||||||
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Passive solar shading is different depending on building orientation, window area distribution between the different faces of the building, and climate. It also requires the presence of an internal thermal mass that can be heated by the direct sun. This thermal mass can be concrete, stone, water, or other material that can store heat. The thermal mass not only stores heat but reduces temperature swings throughout the day."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
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"bbox": [
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "These techniques use the predictable path of the sun (determined through the use of solar position calculators), materials, geometry, and natural environmental conditions to maintain comfort and reduce energy consumption. However, the typical calculations make use of the angle of the sun at solar noon on the Summer and Winter Solstices to calculate the optimal extension of a shade over a window as shown in Figure 1. This is a simplistic view of the problem, and future metrics must do better to account for change."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "image",
|
||||||
|
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|
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"angle": 0,
|
||||||
|
"content": "Figure 1: Passive Solar Shading - Winter and Summer Sun on Solstices"
|
||||||
|
},
|
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|
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"content": "You have been hired by the Collective Organizations Making Astrophysical Protections (COMAP) to innovate the next generation of solar shading strategies to be implemented at the notional Sungrove University and notional Borealis University."
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"angle": 0,
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"content": "The notional Sungrove University, located in a warm, low-latitude region with high solar exposure and increasingly frequent heat waves, is planning a major transformation of its main academic quad. The campus currently suffers from excessive cooling costs and glare in the classrooms. The university leadership has decided to pursue a net-zero cooling initiative by 2040."
|
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|
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],
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"angle": 0,
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"content": "Notably, Sungrove University is planning to retrofit its Academic Hall North. It is a two-story classroom and office building. The interior layout combines perimeter offices and classrooms with interior corridors. The building has a rectangular footprint (60m × 24m) with its long side aligned east-west as shown in Figure 2. The facade consists of double glazing and a brick veneer with an average window-to-wall ratio of \\(45\\%\\) on the south facing side and \\(30\\%\\) on the remaining sides. The building relies on mechanical cooling in the summer and hydronic heating in the winter with limited passive strategies in place. Additional features of this notional building are yours to imagine. Ensure you communicate these features in your writing to COMAP."
|
||||||
|
},
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|
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"type": "image",
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],
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"angle": 0,
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"content": "Figure 2: Academic Hall North footprint"
|
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|
},
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|
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"type": "text",
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"angle": 0,
|
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|
"content": "Additionally, COMAP has been hired by the notional Borealis University, located at a high latitude where winter temperatures remain below freezing for months, sunlight hours are limited, and buildings experience heavy heating demands."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
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],
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"angle": 0,
|
||||||
|
"content": "Sungrove University and Borealis University are also both planning a new student union that will serve as the hub of university activities. They have each mandated that their new student union building relies heavily on passive solar shading rather than mechanical cooling systems. The Universities want their student union building to serve as a prototype for future developments, meaning that their passive solar strategy design must perform well not only today, but under projected climate conditions well into the future."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
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||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Beyond the standard approach to shading as outlined in the Background, to assist these notional universities, you should extend your ideas to include:"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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|
||||||
|
0.841
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Shading needs throughout the day rather than just at solar noon."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
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||||||
|
0.86
|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Windows of different sizes and shapes."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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|
||||||
|
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Windows that do not face exactly south/north (depending on the hemisphere)."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Shades of different styles and materials that would match the architecture of the building."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
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|
"angle": 0,
|
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|
"content": null
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "footer",
|
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|
"bbox": [
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "©2026 by COMAP | www.mathmodels.org | info@comap.org |"
|
||||||
|
}
|
||||||
|
],
|
||||||
|
[
|
||||||
|
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|
||||||
|
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],
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|
"angle": 0,
|
||||||
|
"content": "As with any new strategy or model, you will not only need to describe your approach but also explain the advantages that your proposal holds over the previous standard. COMAP needs to know how your passive solar shading strategies can more effectively reduce heat gain in campus buildings during the summer while still admitting beneficial winter sun."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "title",
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Requirements"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Your team has been asked by COMAP to provide a model-based feasibility analysis that determines how Sungrove University can reduce its academic year cooling load with passive solar design in the retrofit of buildings on campus. To do so, design a retrofit for Sungrove University's Academic Hall North that optimizes heating and cooling throughout the academic year. What passive solar strategies and building features would you use, and how would you evaluate their performance?"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
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|
||||||
|
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Borealis University has a building with a similar design to Sungrove University's Academic Hall North. How can extending your work for Sungrove University to include the crucial importance of the effective use of a thermal mass provide Borealis University with a plan to use passive solar shading? You may want to consider building geometry, material selection, glazing positioning, internal thermal mass, or other aspects to maximize winter heat gain while avoiding overheating in the warmer months."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
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|
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||||||
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|
||||||
|
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "The retrofit design models at both Sungrove and Borealis Universities are helpful for only those notional sites. Adapt your model and discuss the design considerations for other locations including the different heating and cooling needs at places that might have similar latitudes."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
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|
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|
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|
||||||
|
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|
||||||
|
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Design a passive solar shading strategy for the new student union building at either Sungrove University or Borealis University that keeps the building temperate. Describe the strategies, building features, and modeling approaches you would use to evaluate performance over time. You may wish to address some of the following in your analysis:"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
|
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|
||||||
|
0.378,
|
||||||
|
0.605
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Predicting solar heat gain"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
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|
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|
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|
||||||
|
0.57,
|
||||||
|
0.623
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Estimating heating and/or cooling load reductions"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
||||||
|
0.142,
|
||||||
|
0.624,
|
||||||
|
0.451,
|
||||||
|
0.64
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Accounting for seasonal variations"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
||||||
|
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||||||
|
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|
||||||
|
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|
||||||
|
0.659
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Evaluating the tradeoffs between daylighting needs and shading effectiveness"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
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|
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": null
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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||||||
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Write a one-to-two-page letter to either Sungrove University or Borealis University (not both) outlining the steps they should take to include passive solar shading in both their retrofit and new building plans."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Your PDF solution of no more than 25 total pages should include:"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
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"bbox": [
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|
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|
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "One-page Summary Sheet."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Table of Contents."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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|
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Your complete solution."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "One-to-Two-Page Letter."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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],
|
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|
"angle": 0,
|
||||||
|
"content": "- References List."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
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],
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|
"angle": 0,
|
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|
"content": "AI Use Report (If used does not count toward the 25-page limit.)"
|
||||||
|
},
|
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|
{
|
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|
"type": "list",
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"angle": 0,
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|
"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org"
|
||||||
|
}
|
||||||
|
],
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||||||
|
[
|
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|
{
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|
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],
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"angle": 0,
|
||||||
|
"content": "Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution."
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "title",
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"bbox": [
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],
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|
"angle": 0,
|
||||||
|
"content": "Glossary"
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
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"bbox": [
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],
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|
"angle": 0,
|
||||||
|
"content": "Solar noon is the moment during the day when the Sun is at its highest point in the sky for a given location."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
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"angle": 0,
|
||||||
|
"content": "Winter Solstice is the day with the least daylight of the year, caused by Earth's tilt."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
||||||
|
0.112,
|
||||||
|
0.347,
|
||||||
|
0.785,
|
||||||
|
0.365
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Summer Solstice is the day with the most daylight of the year, caused by Earth's tilt."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
||||||
|
0.112,
|
||||||
|
0.374,
|
||||||
|
0.86,
|
||||||
|
0.408
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Notional means theoretical or fictitious. The universities in this problem are not real, but only theoretical case studies."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
||||||
|
0.112,
|
||||||
|
0.419,
|
||||||
|
0.867,
|
||||||
|
0.438
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Net-zero cooling means providing cooling without adding greenhouse gases to the atmosphere."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "footer",
|
||||||
|
"bbox": [
|
||||||
|
0.227,
|
||||||
|
0.939,
|
||||||
|
0.769,
|
||||||
|
0.957
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org"
|
||||||
|
}
|
||||||
|
]
|
||||||
|
]
|
||||||
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|
|||||||
|
# 2026 ICM Problem E: Passive Solar Shading
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
# Background
|
||||||
|
|
||||||
|
Passive solar shading has become a common addition to both housing and commercial buildings as a part of a retrofit or in new construction. It is relatively inexpensive and creates cost savings in heating and cooling. The shades are designed to block summer sun from entering a building, while allowing winter sun to not only enter the building but to warm a thermal mass that can reradiate for many hours after. Strategies such as overhangs, vegetative shading, brise-soleil systems, and high-performance glazing can reduce heat gain in buildings during higher temperatures.
|
||||||
|
|
||||||
|
Passive solar shading is different depending on building orientation, window area distribution between the different faces of the building, and climate. It also requires the presence of an internal thermal mass that can be heated by the direct sun. This thermal mass can be concrete, stone, water, or other material that can store heat. The thermal mass not only stores heat but reduces temperature swings throughout the day.
|
||||||
|
|
||||||
|
These techniques use the predictable path of the sun (determined through the use of solar position calculators), materials, geometry, and natural environmental conditions to maintain comfort and reduce energy consumption. However, the typical calculations make use of the angle of the sun at solar noon on the Summer and Winter Solstices to calculate the optimal extension of a shade over a window as shown in Figure 1. This is a simplistic view of the problem, and future metrics must do better to account for change.
|
||||||
|
|
||||||
|
|
||||||
|
Figure 1: Passive Solar Shading - Winter and Summer Sun on Solstices
|
||||||
|
|
||||||
|
# Scenario
|
||||||
|
|
||||||
|
You have been hired by the Collective Organizations Making Astrophysical Protections (COMAP) to innovate the next generation of solar shading strategies to be implemented at the notional Sungrove University and notional Borealis University.
|
||||||
|
|
||||||
|
The notional Sungrove University, located in a warm, low-latitude region with high solar exposure and increasingly frequent heat waves, is planning a major transformation of its main academic quad. The campus currently suffers from excessive cooling costs and glare in the classrooms. The university leadership has decided to pursue a net-zero cooling initiative by 2040.
|
||||||
|
|
||||||
|
Notably, Sungrove University is planning to retrofit its Academic Hall North. It is a two-story classroom and office building. The interior layout combines perimeter offices and classrooms with interior corridors. The building has a rectangular footprint (60m × 24m) with its long side aligned east-west as shown in Figure 2. The facade consists of double glazing and a brick veneer with an average window-to-wall ratio of $45\%$ on the south facing side and $30\%$ on the remaining sides. The building relies on mechanical cooling in the summer and hydronic heating in the winter with limited passive strategies in place. Additional features of this notional building are yours to imagine. Ensure you communicate these features in your writing to COMAP.
|
||||||
|
|
||||||
|
|
||||||
|
Figure 2: Academic Hall North footprint
|
||||||
|
|
||||||
|
Additionally, COMAP has been hired by the notional Borealis University, located at a high latitude where winter temperatures remain below freezing for months, sunlight hours are limited, and buildings experience heavy heating demands.
|
||||||
|
|
||||||
|
Sungrove University and Borealis University are also both planning a new student union that will serve as the hub of university activities. They have each mandated that their new student union building relies heavily on passive solar shading rather than mechanical cooling systems. The Universities want their student union building to serve as a prototype for future developments, meaning that their passive solar strategy design must perform well not only today, but under projected climate conditions well into the future.
|
||||||
|
|
||||||
|
Beyond the standard approach to shading as outlined in the Background, to assist these notional universities, you should extend your ideas to include:
|
||||||
|
|
||||||
|
- Shading needs throughout the day rather than just at solar noon.
|
||||||
|
- Windows of different sizes and shapes.
|
||||||
|
- Windows that do not face exactly south/north (depending on the hemisphere).
|
||||||
|
- Shades of different styles and materials that would match the architecture of the building.
|
||||||
|
|
||||||
|
As with any new strategy or model, you will not only need to describe your approach but also explain the advantages that your proposal holds over the previous standard. COMAP needs to know how your passive solar shading strategies can more effectively reduce heat gain in campus buildings during the summer while still admitting beneficial winter sun.
|
||||||
|
|
||||||
|
# Requirements
|
||||||
|
|
||||||
|
Your team has been asked by COMAP to provide a model-based feasibility analysis that determines how Sungrove University can reduce its academic year cooling load with passive solar design in the retrofit of buildings on campus. To do so, design a retrofit for Sungrove University's Academic Hall North that optimizes heating and cooling throughout the academic year. What passive solar strategies and building features would you use, and how would you evaluate their performance?
|
||||||
|
|
||||||
|
Borealis University has a building with a similar design to Sungrove University's Academic Hall North. How can extending your work for Sungrove University to include the crucial importance of the effective use of a thermal mass provide Borealis University with a plan to use passive solar shading? You may want to consider building geometry, material selection, glazing positioning, internal thermal mass, or other aspects to maximize winter heat gain while avoiding overheating in the warmer months.
|
||||||
|
|
||||||
|
The retrofit design models at both Sungrove and Borealis Universities are helpful for only those notional sites. Adapt your model and discuss the design considerations for other locations including the different heating and cooling needs at places that might have similar latitudes.
|
||||||
|
|
||||||
|
Design a passive solar shading strategy for the new student union building at either Sungrove University or Borealis University that keeps the building temperate. Describe the strategies, building features, and modeling approaches you would use to evaluate performance over time. You may wish to address some of the following in your analysis:
|
||||||
|
|
||||||
|
- Predicting solar heat gain
|
||||||
|
- Estimating heating and/or cooling load reductions
|
||||||
|
- Accounting for seasonal variations
|
||||||
|
- Evaluating the tradeoffs between daylighting needs and shading effectiveness
|
||||||
|
|
||||||
|
Write a one-to-two-page letter to either Sungrove University or Borealis University (not both) outlining the steps they should take to include passive solar shading in both their retrofit and new building plans.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One-to-Two-Page Letter.
|
||||||
|
- References List.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
Solar noon is the moment during the day when the Sun is at its highest point in the sky for a given location.
|
||||||
|
|
||||||
|
Winter Solstice is the day with the least daylight of the year, caused by Earth's tilt.
|
||||||
|
|
||||||
|
Summer Solstice is the day with the most daylight of the year, caused by Earth's tilt.
|
||||||
|
|
||||||
|
Notional means theoretical or fictitious. The universities in this problem are not real, but only theoretical case studies.
|
||||||
|
|
||||||
|
Net-zero cooling means providing cooling without adding greenhouse gases to the atmosphere.
|
||||||
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|
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|
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|
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|
|||||||
|
[
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "2026 ICM",
|
||||||
|
"text_level": 1,
|
||||||
|
"bbox": [
|
||||||
|
455,
|
||||||
|
90,
|
||||||
|
542,
|
||||||
|
106
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Problem F: To Gen-AI, or Not To Gen-AI (or how to Gen-AI)? That is the Question!",
|
||||||
|
"text_level": 1,
|
||||||
|
"bbox": [
|
||||||
|
143,
|
||||||
|
109,
|
||||||
|
854,
|
||||||
|
125
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "image",
|
||||||
|
"img_path": "images/e22c5822463cf6f31feee50f4cc9573edbf6d99fa928c613ab5390964ee32cd6.jpg",
|
||||||
|
"image_caption": [],
|
||||||
|
"image_footnote": [],
|
||||||
|
"bbox": [
|
||||||
|
326,
|
||||||
|
140,
|
||||||
|
669,
|
||||||
|
319
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "In just a few years, Generative Artificial Intelligence (Gen-AI) has gone from a tool of limited capacity, only used by a few early-adopters to a powerful and inescapable resource embedded in our daily lives. Over time, research suggests that Gen-AI could impact the future of work. For example, in some fields, Gen-AI may replace humans (or heavily reduce the human workload), while other fields might not be heavily impacted or might even grow.",
|
||||||
|
"bbox": [
|
||||||
|
109,
|
||||||
|
345,
|
||||||
|
874,
|
||||||
|
434
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "In this question, you will explore how post-secondary educational institutions, of various types, should best prepare their future graduates in light of this new technology. Specifically, you are asking do to the following.",
|
||||||
|
"bbox": [
|
||||||
|
109,
|
||||||
|
444,
|
||||||
|
867,
|
||||||
|
497
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "- Choose three careers, one from each of the following categories:",
|
||||||
|
"bbox": [
|
||||||
|
142,
|
||||||
|
506,
|
||||||
|
683,
|
||||||
|
521
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
"sub_type": "text",
|
||||||
|
"list_items": [
|
||||||
|
"STEM career: people in this career often have at least a four year university degree in the sciences, engineering, or mathematics;",
|
||||||
|
"Trade career: people in this career often have training from a trade school and/or an apprenticeship program, such as chef, plumber, and electrician;",
|
||||||
|
"○ Arts career: people in this career often have studied at an arts school, conservatory, or cultural center, such as musician, dancer, or painter."
|
||||||
|
],
|
||||||
|
"bbox": [
|
||||||
|
200,
|
||||||
|
523,
|
||||||
|
866,
|
||||||
|
625
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "- Design a data-informed model to explore the future of each of your three chosen professions, given the current trajectory and expected impacts of Gen-AI. Be sure to identify your data sources as well as your reasoning behind any drivers that you expect to change this profession as a result of Gen-AI. Note: you may leverage existing research on the future of work, but be sure to cite your sources and explain how you are using the established research to inform your analysis.",
|
||||||
|
"bbox": [
|
||||||
|
142,
|
||||||
|
628,
|
||||||
|
880,
|
||||||
|
732
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "- Identify a specific post-secondary institution and program of study for each career you are analyzing (one at a university, one at a trade school, one at an arts school), and focus your recommendations accordingly. In other words, you should have three sets of recommendations that address the following question: Based on your analysis, how would you advise the leaders of each of these institutions to address Gen-AI in the programs specific to the careers you are analyzing?",
|
||||||
|
"bbox": [
|
||||||
|
142,
|
||||||
|
733,
|
||||||
|
870,
|
||||||
|
835
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Below are just some thoughts you may wish to consider; teams should not attempt to address all of these ideas, but should use these as inspiration that will lead to a cogent and thorough analysis that should vary team to team.",
|
||||||
|
"bbox": [
|
||||||
|
169,
|
||||||
|
838,
|
||||||
|
880,
|
||||||
|
888
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "footer",
|
||||||
|
"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org",
|
||||||
|
"bbox": [
|
||||||
|
223,
|
||||||
|
938,
|
||||||
|
767,
|
||||||
|
955
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "○ Should the program of study grow or shrink (graduate more or fewer people) as a result of changes in the career due to Gen-AI? If the field should grow, how might the institution recruit more people; and if the field should shrink, are there other programs in the school that should grow to absorb the people who used to study in this program?",
|
||||||
|
"bbox": [
|
||||||
|
200,
|
||||||
|
90,
|
||||||
|
879,
|
||||||
|
176
|
||||||
|
],
|
||||||
|
"page_idx": 1
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "○ What should these three different programs of study teach about Gen-AI? Many post-secondary institutions of learning have asked this question and are still developing their response. While some institutions have outright banned the use of AI on any assignments, others have brought the use of AI to the forefront of their curriculum. Some schools aim to produce experts who can contribute to the leading edge of the technological field, while some focus on graduating students in non-technical fields who are fluent users of the technology. Some institutions encourage their students to think about all the ways they can apply this new technology, and some schools challenge students to carefully weigh the benefits and costs of using AI, given the requisite energy usage, water demands, and risk of insufficient (often missing or incorrect) attribution to the original creators of ideas or content. For the three programs of study at the three institutions you've selected, what do you recommend to best support the employability of their graduates? Be sure to support your recommendations with the results of a mathematical model.",
|
||||||
|
"bbox": [
|
||||||
|
200,
|
||||||
|
178,
|
||||||
|
867,
|
||||||
|
435
|
||||||
|
],
|
||||||
|
"page_idx": 1
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "While this problem poses the question through the context of employability of graduates in a world where Gen-AI is ubiquitous, perhaps employment demands are not the only way to measure the success of the institutional policies you are proposing. What other factors do you believe should be considered, and how do your models and recommendations change when you consider these other factors?",
|
||||||
|
"bbox": [
|
||||||
|
200,
|
||||||
|
439,
|
||||||
|
879,
|
||||||
|
525
|
||||||
|
],
|
||||||
|
"page_idx": 1
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "If you believe your specific recommendations can be generalized beyond one institution and/or beyond one program, be sure to explain the extent of the generalization and justify this.",
|
||||||
|
"bbox": [
|
||||||
|
200,
|
||||||
|
527,
|
||||||
|
841,
|
||||||
|
578
|
||||||
|
],
|
||||||
|
"page_idx": 1
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Your PDF solution of no more than 25 total pages should include:",
|
||||||
|
"bbox": [
|
||||||
|
112,
|
||||||
|
614,
|
||||||
|
594,
|
||||||
|
633
|
||||||
|
],
|
||||||
|
"page_idx": 1
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
"sub_type": "text",
|
||||||
|
"list_items": [
|
||||||
|
"One-page Summary Sheet.",
|
||||||
|
"Table of Contents.",
|
||||||
|
"- Your complete solution.",
|
||||||
|
"- References list.",
|
||||||
|
"AI Use Report (If used does not count toward the 25-page limit.)"
|
||||||
|
],
|
||||||
|
"bbox": [
|
||||||
|
140,
|
||||||
|
651,
|
||||||
|
684,
|
||||||
|
751
|
||||||
|
],
|
||||||
|
"page_idx": 1
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.",
|
||||||
|
"bbox": [
|
||||||
|
109,
|
||||||
|
768,
|
||||||
|
880,
|
||||||
|
898
|
||||||
|
],
|
||||||
|
"page_idx": 1
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "footer",
|
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"content": "In this question, you will explore how post-secondary educational institutions, of various types, should best prepare their future graduates in light of this new technology. Specifically, you are asking do to the following."
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"content": "- Choose three careers, one from each of the following categories:"
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"content": "STEM career: people in this career often have at least a four year university degree in the sciences, engineering, or mathematics;"
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"content": "Trade career: people in this career often have training from a trade school and/or an apprenticeship program, such as chef, plumber, and electrician;"
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"content": "○ Arts career: people in this career often have studied at an arts school, conservatory, or cultural center, such as musician, dancer, or painter."
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"angle": 0,
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"content": "- Design a data-informed model to explore the future of each of your three chosen professions, given the current trajectory and expected impacts of Gen-AI. Be sure to identify your data sources as well as your reasoning behind any drivers that you expect to change this profession as a result of Gen-AI. Note: you may leverage existing research on the future of work, but be sure to cite your sources and explain how you are using the established research to inform your analysis."
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"angle": 0,
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"content": "- Identify a specific post-secondary institution and program of study for each career you are analyzing (one at a university, one at a trade school, one at an arts school), and focus your recommendations accordingly. In other words, you should have three sets of recommendations that address the following question: Based on your analysis, how would you advise the leaders of each of these institutions to address Gen-AI in the programs specific to the careers you are analyzing?"
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"content": "Below are just some thoughts you may wish to consider; teams should not attempt to address all of these ideas, but should use these as inspiration that will lead to a cogent and thorough analysis that should vary team to team."
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"content": "○ Should the program of study grow or shrink (graduate more or fewer people) as a result of changes in the career due to Gen-AI? If the field should grow, how might the institution recruit more people; and if the field should shrink, are there other programs in the school that should grow to absorb the people who used to study in this program?"
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"content": "○ What should these three different programs of study teach about Gen-AI? Many post-secondary institutions of learning have asked this question and are still developing their response. While some institutions have outright banned the use of AI on any assignments, others have brought the use of AI to the forefront of their curriculum. Some schools aim to produce experts who can contribute to the leading edge of the technological field, while some focus on graduating students in non-technical fields who are fluent users of the technology. Some institutions encourage their students to think about all the ways they can apply this new technology, and some schools challenge students to carefully weigh the benefits and costs of using AI, given the requisite energy usage, water demands, and risk of insufficient (often missing or incorrect) attribution to the original creators of ideas or content. For the three programs of study at the three institutions you've selected, what do you recommend to best support the employability of their graduates? Be sure to support your recommendations with the results of a mathematical model."
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"content": "While this problem poses the question through the context of employability of graduates in a world where Gen-AI is ubiquitous, perhaps employment demands are not the only way to measure the success of the institutional policies you are proposing. What other factors do you believe should be considered, and how do your models and recommendations change when you consider these other factors?"
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"content": "If you believe your specific recommendations can be generalized beyond one institution and/or beyond one program, be sure to explain the extent of the generalization and justify this."
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"angle": 0,
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"content": "Your PDF solution of no more than 25 total pages should include:"
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"angle": 0,
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"angle": 0,
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"content": "- Your complete solution."
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"content": "- References list."
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@@ -0,0 +1,39 @@
|
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|
# 2026 ICM
|
||||||
|
|
||||||
|
# Problem F: To Gen-AI, or Not To Gen-AI (or how to Gen-AI)? That is the Question!
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
In just a few years, Generative Artificial Intelligence (Gen-AI) has gone from a tool of limited capacity, only used by a few early-adopters to a powerful and inescapable resource embedded in our daily lives. Over time, research suggests that Gen-AI could impact the future of work. For example, in some fields, Gen-AI may replace humans (or heavily reduce the human workload), while other fields might not be heavily impacted or might even grow.
|
||||||
|
|
||||||
|
In this question, you will explore how post-secondary educational institutions, of various types, should best prepare their future graduates in light of this new technology. Specifically, you are asking do to the following.
|
||||||
|
|
||||||
|
- Choose three careers, one from each of the following categories:
|
||||||
|
|
||||||
|
STEM career: people in this career often have at least a four year university degree in the sciences, engineering, or mathematics;
|
||||||
|
Trade career: people in this career often have training from a trade school and/or an apprenticeship program, such as chef, plumber, and electrician;
|
||||||
|
○ Arts career: people in this career often have studied at an arts school, conservatory, or cultural center, such as musician, dancer, or painter.
|
||||||
|
|
||||||
|
- Design a data-informed model to explore the future of each of your three chosen professions, given the current trajectory and expected impacts of Gen-AI. Be sure to identify your data sources as well as your reasoning behind any drivers that you expect to change this profession as a result of Gen-AI. Note: you may leverage existing research on the future of work, but be sure to cite your sources and explain how you are using the established research to inform your analysis.
|
||||||
|
|
||||||
|
- Identify a specific post-secondary institution and program of study for each career you are analyzing (one at a university, one at a trade school, one at an arts school), and focus your recommendations accordingly. In other words, you should have three sets of recommendations that address the following question: Based on your analysis, how would you advise the leaders of each of these institutions to address Gen-AI in the programs specific to the careers you are analyzing?
|
||||||
|
|
||||||
|
Below are just some thoughts you may wish to consider; teams should not attempt to address all of these ideas, but should use these as inspiration that will lead to a cogent and thorough analysis that should vary team to team.
|
||||||
|
|
||||||
|
○ Should the program of study grow or shrink (graduate more or fewer people) as a result of changes in the career due to Gen-AI? If the field should grow, how might the institution recruit more people; and if the field should shrink, are there other programs in the school that should grow to absorb the people who used to study in this program?
|
||||||
|
|
||||||
|
○ What should these three different programs of study teach about Gen-AI? Many post-secondary institutions of learning have asked this question and are still developing their response. While some institutions have outright banned the use of AI on any assignments, others have brought the use of AI to the forefront of their curriculum. Some schools aim to produce experts who can contribute to the leading edge of the technological field, while some focus on graduating students in non-technical fields who are fluent users of the technology. Some institutions encourage their students to think about all the ways they can apply this new technology, and some schools challenge students to carefully weigh the benefits and costs of using AI, given the requisite energy usage, water demands, and risk of insufficient (often missing or incorrect) attribution to the original creators of ideas or content. For the three programs of study at the three institutions you've selected, what do you recommend to best support the employability of their graduates? Be sure to support your recommendations with the results of a mathematical model.
|
||||||
|
|
||||||
|
While this problem poses the question through the context of employability of graduates in a world where Gen-AI is ubiquitous, perhaps employment demands are not the only way to measure the success of the institutional policies you are proposing. What other factors do you believe should be considered, and how do your models and recommendations change when you consider these other factors?
|
||||||
|
|
||||||
|
If you believe your specific recommendations can be generalized beyond one institution and/or beyond one program, be sure to explain the extent of the generalization and justify this.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
- References list.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
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"content": "In just a few years, Generative Artificial Intelligence (Gen-AI) has gone from a tool of limited capacity, only used by a few early-adopters to a powerful and inescapable resource embedded in our daily lives. Over time, research suggests that Gen-AI could impact the future of work. For example, in some fields, Gen-AI may replace humans (or heavily reduce the human workload), while other fields might not be heavily impacted or might even grow."
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"type": "text",
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"content": "In this question, you will explore how post-secondary educational institutions, of various types, should best prepare their future graduates in light of this new technology. Specifically, you are asking do to the following."
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"type": "text",
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"content": "- Choose three careers, one from each of the following categories:"
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|
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|
441
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|
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|
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|
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|
||||||
|
"type": "text",
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||||||
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"content": "STEM career: people in this career often have at least a four year university degree in the sciences, engineering, or mathematics;"
|
||||||
|
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|
||||||
|
]
|
||||||
|
}
|
||||||
|
],
|
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|
"index": 6
|
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|
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|
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|
{
|
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|
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|
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|
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|
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|
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|
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|
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|
468
|
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|
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|
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|
"type": "text",
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
"type": "text",
|
||||||
|
"content": "Trade career: people in this career often have training from a trade school and/or an apprenticeship program, such as chef, plumber, and electrician;"
|
||||||
|
}
|
||||||
|
]
|
||||||
|
}
|
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|
],
|
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|
"index": 7
|
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|
},
|
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|
{
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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"type": "text",
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"content": "○ Arts career: people in this career often have studied at an arts school, conservatory, or cultural center, such as musician, dancer, or painter."
|
||||||
|
}
|
||||||
|
]
|
||||||
|
}
|
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|
],
|
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|
"index": 8
|
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|
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|
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|
],
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|
"sub_type": "text"
|
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|
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|
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|
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|
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|
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|
"type": "text",
|
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|
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|
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|
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|
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|
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|
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|
"type": "text",
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|
"content": "- Design a data-informed model to explore the future of each of your three chosen professions, given the current trajectory and expected impacts of Gen-AI. Be sure to identify your data sources as well as your reasoning behind any drivers that you expect to change this profession as a result of Gen-AI. Note: you may leverage existing research on the future of work, but be sure to cite your sources and explain how you are using the established research to inform your analysis."
|
||||||
|
}
|
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|
]
|
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|
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|
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|
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|
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|
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|
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|
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|
"type": "text",
|
||||||
|
"content": "- Identify a specific post-secondary institution and program of study for each career you are analyzing (one at a university, one at a trade school, one at an arts school), and focus your recommendations accordingly. In other words, you should have three sets of recommendations that address the following question: Based on your analysis, how would you advise the leaders of each of these institutions to address Gen-AI in the programs specific to the careers you are analyzing?"
|
||||||
|
}
|
||||||
|
]
|
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|
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|
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|
],
|
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|
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|
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"type": "text",
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"content": "Below are just some thoughts you may wish to consider; teams should not attempt to address all of these ideas, but should use these as inspiration that will lead to a cogent and thorough analysis that should vary team to team."
|
||||||
|
}
|
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|
]
|
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|
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|
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"type": "text",
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"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org"
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]
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"type": "text",
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"content": "○ Should the program of study grow or shrink (graduate more or fewer people) as a result of changes in the career due to Gen-AI? If the field should grow, how might the institution recruit more people; and if the field should shrink, are there other programs in the school that should grow to absorb the people who used to study in this program?"
|
||||||
|
}
|
||||||
|
]
|
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|
}
|
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BIN
2026_MCM-ICM_Problems/2026_ICM_Problem_D.pdf
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2026_MCM-ICM_Problems/2026_ICM_Problem_D.pdf
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2026_MCM-ICM_Problems/2026_ICM_Problem_E.pdf
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2026_MCM-ICM_Problems/2026_ICM_Problem_E.pdf
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2026_MCM-ICM_Problems/2026_ICM_Problem_F.pdf
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2026_MCM-ICM_Problems/2026_ICM_Problem_F.pdf
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2026_MCM-ICM_Problems/2026_MCM_Problem_A.pdf
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2026_MCM-ICM_Problems/2026_MCM_Problem_A.pdf
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2026_MCM-ICM_Problems/2026_MCM_Problem_B.pdf
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2026_MCM-ICM_Problems/2026_MCM_Problem_B.pdf
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2026_MCM-ICM_Problems/2026_MCM_Problem_C.pdf
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2026_MCM-ICM_Problems/2026_MCM_Problem_C.pdf
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422
2026_MCM-ICM_Problems/2026_MCM_Problem_C_Data.csv
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422
2026_MCM-ICM_Problems/2026_MCM_Problem_C_Data.csv
Normal file
@@ -0,0 +1,422 @@
|
|||||||
|
celebrity_name,ballroom_partner,celebrity_industry,celebrity_homestate,celebrity_homecountry/region,celebrity_age_during_season,season,results,placement,week1_judge1_score,week1_judge2_score,week1_judge3_score,week1_judge4_score,week2_judge1_score,week2_judge2_score,week2_judge3_score,week2_judge4_score,week3_judge1_score,week3_judge2_score,week3_judge3_score,week3_judge4_score,week4_judge1_score,week4_judge2_score,week4_judge3_score,week4_judge4_score,week5_judge1_score,week5_judge2_score,week5_judge3_score,week5_judge4_score,week6_judge1_score,week6_judge2_score,week6_judge3_score,week6_judge4_score,week7_judge1_score,week7_judge2_score,week7_judge3_score,week7_judge4_score,week8_judge1_score,week8_judge2_score,week8_judge3_score,week8_judge4_score,week9_judge1_score,week9_judge2_score,week9_judge3_score,week9_judge4_score,week10_judge1_score,week10_judge2_score,week10_judge3_score,week10_judge4_score,week11_judge1_score,week11_judge2_score,week11_judge3_score,week11_judge4_score
|
||||||
|
John O'Hurley,Charlotte Jorgensen,Actor/Actress,Maine,United States,50,1,2nd Place,2,7,7,6,N/A,8,9,9,N/A,9,8,7,N/A,7,8,6,N/A,9,9,9,N/A,9,9,9,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kelly Monaco,Alec Mazo,Actor/Actress,Pennsylvania,United States,29,1,1st Place,1,5,4,4,N/A,5,6,6,N/A,6,7,8,N/A,9,9,8,N/A,8.5,7.5,7.5,N/A,8.5,9.5,9.5,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Evander Holyfield,Edyta Sliwinska,Athlete,Alabama,United States,42,1,Eliminated Week 3,5,5,7,6,N/A,5,4,5,N/A,5,4,4,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Rachel Hunter,Jonathan Roberts,Model,,New Zealand,35,1,Eliminated Week 4,4,7,6,7,N/A,8,8,8,N/A,8,9,9,N/A,7,9,9,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Joey McIntyre,Ashly DelGrosso,Singer/Rapper,Massachusetts,United States,32,1,3rd Place,3,7,7,6,N/A,8,7,6,N/A,7,7,8,N/A,7,6,7,N/A,8.5,7,7,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Trista Sutter,Louis van Amstel,TV Personality,Indiana,United States,32,1,Eliminated Week 2,6,6,6,6,N/A,6,7,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Tatum O'Neal,Nick Kosovich,Actor/Actress,California,United States,42,2,Eliminated Week 2,9,7,8,8,N/A,5,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Tia Carrere,Maksim Chmerkoskiy,Actor/Actress,Hawaii,United States,39,2,Eliminated Week 5,6,6,7,7,N/A,7,8,7,N/A,9,8,9,N/A,9,8,8,N/A,7,7,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
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|
George Hamilton,Edyta Sliwinska,Actor/Actress,Tennessee,United States,66,2,Eliminated Week 6,5,7,5,6,N/A,8,7,7,N/A,7,7,8,N/A,7,7,7,N/A,8,8,8,N/A,8,7,8,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
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|
Lisa Rinna,Louis van Amstel,Actor/Actress,California,United States,42,2,Eliminated Week 7,4,5,7,7,N/A,6,7,7,N/A,8,8,9,N/A,9,9,8,N/A,7,9,9,N/A,9,9,9,N/A,8.5,9,9,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Stacy Keibler,Tony Dovolani,Athlete,Maryland,United States,26,2,3rd Place,3,8,6,8,N/A,9,10,10,N/A,9,9,9,N/A,8,9,9,N/A,10,10,10,N/A,10,10,10,N/A,9,9,9.5,N/A,9.3333,9.6666,9.6666,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jerry Rice,Anna Trebunskaya,Athlete,Mississippi,United States,43,2,2nd Place,2,7,7,7,N/A,7,8,8,N/A,7,6,6,N/A,8,8,8,N/A,7,8,8,N/A,8,7,8,N/A,7,7,6.5,N/A,9,9,8.6666,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Giselle Fernandez,Jonathan Roberts,News Anchor,,Mexico,44,2,Eliminated Week 3,8,7,8,8,N/A,8,8,8,N/A,7,8,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
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|
Master P,Ashly DelGrosso,Singer/Rapper,Louisiana,United States,35,2,Eliminated Week 4,7,4,4,4,N/A,6,5,5,N/A,6,4,4,N/A,4,2,2,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Drew Lachey,Cheryl Burke,Singer/Rapper,Ohio,United States,29,2,1st Place,1,8,8,8,N/A,9,9,9,N/A,9,9,9,N/A,9,9,10,N/A,9,9,9,N/A,10,10,10,N/A,9.5,9,9,N/A,9.6666,9.6666,9.6666,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kenny Mayne,Andrea Hale,Sports Broadcaster,Washington,United States,46,2,Eliminated Week 1,10,4,5,4,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Harry Hamlin,Ashly DelGrosso,Actor/Actress,California,United States,54,3,Eliminated Week 3,9,5,6,6,N/A,7,7,7,N/A,7,8,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Vivica A. Fox,Nick Kosovich,Actor/Actress,Indiana,United States,42,3,Eliminated Week 4,8,6,8,8,N/A,8,8,8,N/A,9,9,9,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Monique Coleman,Louis van Amstel,Actor/Actress,South Carolina,United States,25,3,Eliminated Week 8,4,6,6,7,N/A,9,8,9,N/A,9,9,9,N/A,8,8,8,N/A,9,9,9,N/A,9,7,7,N/A,9,9,9,N/A,8.5,9,9,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Joey Lawrence,Edyta Sliwinska,Actor/Actress,Pennsylvania,United States,30,3,3rd Place,3,7,7,7,N/A,10,9,10,N/A,8,6,8,N/A,9,9,9,N/A,8,8,9,N/A,8,8,8,N/A,9.5,9,10,N/A,9.5,8.5,9,N/A,9.5,10,10,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Mario Lopez,Karina Smirnoff,Actor/Actress,California,United States,32,3,2nd Place,2,9,8,9,N/A,7,6,8,N/A,8,6,8,N/A,10,9,10,N/A,9,9,9,N/A,9,9,10,N/A,9.5,9,9.5,N/A,9.5,9,10,N/A,10,9.5,10,N/A,10,10,9.6666,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Emmitt Smith,Cheryl Burke,Athlete,Florida,United States,37,3,1st Place,1,8,8,8,N/A,8,8,8,N/A,7,6,6,N/A,8,8,8,N/A,9,9,9,N/A,8,8,9,N/A,10,9.5,9,N/A,8.5,9,9.5,N/A,9.5,10,10,N/A,10,9.6666,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Shanna Moakler,Jesse DeSoto,Model,Rhode Island,United States,31,3,Eliminated Week 2,10,7,5,6,N/A,8,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Willa Ford,Maksim Chmerkoskiy,Singer/Rapper,Florida,United States,25,3,Eliminated Week 5,7,7,7,8,N/A,7,8,8,N/A,7,7,8,N/A,9,9,10,N/A,9,9,9,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Sara Evans,Tony Dovolani,Singer/Rapper,Missouri,United States,35,3,Withdrew,6,5,5,5,N/A,7,7,7,N/A,8,9,8,N/A,6,7,7,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Jerry Springer,Kym Johnson,TV Personality,,England,62,3,Eliminated Week 7,5,5,5,6,N/A,7,6,6,N/A,7,7,7,N/A,7,7,8,N/A,8,8,8,N/A,7,6,5,N/A,7.5,8,7.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Tucker Carlson,Elena Grinenko,TV Personality,California,United States,37,3,Eliminated Week 1,11,5,4,3,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
John Ratzenberger,Edyta Sliwinska,Actor/Actress,Connecticut,United States,59,4,Eliminated Week 7,6,6,5,6,N/A,7,7,7,N/A,7,6,7,N/A,6,5,5,N/A,6,6,6,N/A,7,6,6,N/A,7.5,7.5,7.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Ian Ziering,Cheryl Burke,Actor/Actress,New Jersey,United States,42,4,Eliminated Week 9,4,7,7,7,N/A,7,8,7,N/A,8,8,8,N/A,7,9,8,N/A,8,8,8,N/A,8,8,8,N/A,9,9,9,N/A,8,7.5,8,N/A,9.5,10,9.5,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Clyde Drexler,Elena Grinenko,Athlete,Louisiana,United States,44,4,Eliminated Week 5,8,6,5,5,N/A,6,6,6,N/A,6,5,5,N/A,6,4,5,N/A,4,5,4,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Laila Ali,Maksim Chmerkoskiy,Athlete,Florida,United States,29,4,3rd Place,3,7,8,8,N/A,9,9,9,N/A,7,7,7,N/A,7,7,7,N/A,9,10,9,N/A,9,9,10,N/A,10,9.5,10,N/A,9,8.5,9,N/A,10,10,10,N/A,9.6666,9,9.6666,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Apolo Anton Ohno,Julianne Hough,Athlete,Washington,United States,24,4,1st Place,1,7,7,7,N/A,8,9,9,N/A,7,8,8,N/A,9,8,9,N/A,10,10,10,N/A,9,9,10,N/A,9,8.5,9.5,N/A,10,9,10,N/A,10,9.5,10,N/A,9.6666,9.6666,10,N/A,N/A,N/A,N/A,N/A
|
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|
Shandi Finnessey,Brian Fortuna,Beauty Pagent,Missouri,United States,28,4,Eliminated Week 3,10,6,6,7,N/A,6,7,7,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Paulina Porizkova,Alec Mazo,Model,,Czechoslovakia,41,4,Eliminated Week 2,11,6,6,7,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Heather Mills,Jonathan Roberts,Model,,England,39,4,Eliminated Week 6,7,6,6,6,N/A,8,8,8,N/A,8,8,8,N/A,7,8,8,N/A,7,7,7,N/A,7,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Billy Ray Cyrus,Karina Smirnoff,Singer/Rapper,Kentucky,United States,45,4,Eliminated Week 8,5,5,4,4,N/A,7,7,7,N/A,7,7,7,N/A,7,7,7,N/A,6,6,5,N/A,7,7,7,N/A,6,6.5,6.5,N/A,6.5,6.5,6,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Joey Fatone,Kym Johnson,Singer/Rapper,New York,United States,30,4,2nd Place,2,8,8,8,N/A,8,8,8,N/A,8,8,8,N/A,10,9,9,N/A,8,8,9,N/A,9,9,9,N/A,10,9.5,10,N/A,9.5,9,9,N/A,10,10,10,N/A,9.6666,9.3333,9.6666,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Leeza Gibbons,Tony Dovolani,TV Personality,South Carolina,United States,50,4,Eliminated Week 4,9,5,5,5,N/A,7,7,7,N/A,8,8,8,N/A,6,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Cameron Mathison,Edyta Sliwinska,Actor/Actress,,Canada,38,5,Eliminated Week 8,5,7,7,7,N/A,7,7,7,N/A,8,7,8,N/A,9,9,9,N/A,8,9,9,N/A,9,8,8,N/A,8.5,8.5,8.5,N/A,8.5,8.5,8.5,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Jane Seymour,Tony Dovolani,Actor/Actress,,England,56,5,Eliminated Week 7,6,8,8,8,N/A,7,7,7,N/A,9,9,9,N/A,8,9,9,N/A,8,9,9,N/A,8,7,7,N/A,8,8.5,8.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Sabrina Bryan,Mark Ballas,Actor/Actress,California,United States,23,5,Eliminated Week 6,7,9,8,9,N/A,9,8,9,N/A,9,9,9,N/A,10,10,10,N/A,9,9,10,N/A,9,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jennie Garth,Derek Hough,Actor/Actress,Illinois,United States,35,5,Eliminated Week 9,4,7,7,7,N/A,7,7,7,N/A,9,8,9,N/A,8,10,9,N/A,8,9,8,N/A,9,9,9,N/A,8.5,8.5,9.5,N/A,8.5,8.5,8,N/A,9.5,10,9.5,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Floyd Mayweather Jr. ,Karina Smirnoff,Athlete,Michigan,United States,30,5,Eliminated Week 4,9,6,6,6,N/A,7,7,7,N/A,7,7,7,N/A,7,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Josie Maran,Alec Mazo,Model,California,United States,29,5,Eliminated Week 1,12,6,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Albert Reed,Anna Trebunskaya,Model,Florida,United States,22,5,Eliminated Week 2,11,7,7,7,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Helio Castroneves,Julianne Hough,Racing Driver,,Brazil,32,5,1st Place,1,8,9,8,N/A,9,9,9,N/A,8,8,8,N/A,9,9,9,N/A,8,7,8,N/A,9,10,9,N/A,9,8.5,8.5,N/A,9.5,9.5,9.5,N/A,10,10,10,N/A,9,9.3333,9.6666,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Mel B,Maksim Chmerkoskiy,Singer/Rapper,,England,32,5,2nd Place,2,8,8,8,N/A,7,8,8,N/A,9,9,9,N/A,8,9,9,N/A,10,9,10,N/A,10,10,10,N/A,9,9,9,N/A,9,9.5,9.5,N/A,10,10,10,N/A,9.3333,9.3333,9.6666,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Wayne Newton,Cheryl Burke,Singer/Rapper,Virginia,United States,65,5,Eliminated Week 3,10,6,7,6,N/A,5,5,5,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Marie Osmond,Jonathan Roberts,Singer/Rapper,Utah,United States,47,5,3rd Place,3,7,7,7,N/A,8,8,8,N/A,9,8,9,N/A,9,9,8,N/A,7,7,7,N/A,8,8,7,N/A,9,8.5,8.5,N/A,8,8.5,8,N/A,9.5,9.5,9,N/A,8,7.5,7.5,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Mark Cuban,Kym Johnson,TV Personality,Pennsylvania,United States,49,5,Eliminated Week 5,8,7,7,7,N/A,6,6,6,N/A,6,7,7,N/A,7,8,7,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Cristian de la Fuente,Cheryl Burke,Actor/Actress,,Chile,34,6,3rd Place,3,7,7,7,N/A,7,6,7,N/A,8,8,9,N/A,9,8,9,N/A,7,8,8,N/A,9,9,9,N/A,7.5,7.5,8,N/A,10,9,9.5,N/A,9.5,9,9.5,N/A,9,8,9,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Steve Guttenberg,Anna Trebunskaya,Actor/Actress,New York,United States,49,6,Eliminated Week 3,10,6,6,6,N/A,6,5,5,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Priscilla Presley,Louis van Amstel,Actor/Actress,New York,United States,62,6,Eliminated Week 5,8,8,8,8,N/A,7,7,7,N/A,8,9,9,N/A,7,7,8,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Marlee Matlin,Fabian Sanchez,Actor/Actress,Illinois,United States,42,6,Eliminated Week 6,7,7,7,8,N/A,8,8,8,N/A,7,7,7,N/A,8,8,8,N/A,7,7,8,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Shannon Elizabeth,Derek Hough,Actor/Actress,Texas,United States,34,6,Eliminated Week 7,6,7,7,7,N/A,8,8,8,N/A,8,8,8,N/A,9,10,9,N/A,8,8,7,N/A,8,8,8,N/A,8.5,8.5,8.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Marissa Jaret Winokur,Tony Dovolani,Actor/Actress,New York,United States,35,6,Eliminated Week 9,4,6,6,6,N/A,7,7,7,N/A,6,7,6,N/A,8,8,8,N/A,8,8,8,N/A,9,8,9,N/A,9,8.5,8.5,N/A,8.5,8,8.5,N/A,8.5,9,8.5,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jason Taylor,Edyta Sliwinska,Athlete,Pennsylvania,United States,33,6,2nd Place,2,7,8,7,N/A,9,9,9,N/A,8,7,8,N/A,10,9,10,N/A,9,9,9,N/A,8,8,8,N/A,9.5,8.5,9.5,N/A,9,8,9,N/A,9,9.5,9,N/A,9.5,9.5,9.5,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kristi Yamaguchi,Mark Ballas,Athlete,California,United States,36,6,1st Place,1,9,9,9,N/A,9,9,9,N/A,9,9,9,N/A,10,9,10,N/A,9,10,10,N/A,10,10,10,N/A,9.5,8,9.5,N/A,8.5,9.5,9.5,N/A,9.5,9,10,N/A,10,10,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Monica Seles,Jonathan Roberts,Athlete,,Yugoslavia,34,6,Eliminated Week 2,11,5,5,5,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Penn Jillette,Kym Johnson,Magician,Massachusetts,United States,53,6,Eliminated Week 2,12,5,6,5,N/A,6,6,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Adam Carolla,Julianne Hough,Radio Personality,California,United States,43,6,Eliminated Week 4,9,5,5,5,N/A,6,7,6,N/A,7,7,7,N/A,6,7,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Mario,Karina Smirnoff,Singer/Rapper,Maryland,United States,21,6,Eliminated Week 8,5,8,8,8,N/A,9,8,9,N/A,7,6,8,N/A,8,7,9,N/A,9,9,9,N/A,9,9,10,N/A,8.5,8.5,8.5,N/A,9,8.5,9,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Ted McGinley,Inna Brayer,Actor/Actress,California,United States,50,7,Eliminated Week 1,12,6,6,6.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Cloris Leachman,Corky Ballas,Actor/Actress,Iowa,United States,82,7,Eliminated Week 6,7,6,5,5,N/A,5,5,5,N/A,6,5,5,N/A,8,7,7,N/A,7,7,7,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Susan Lucci,Tony Dovolani,Actor/Actress,New York,United States,61,7,Eliminated Week 7,6,6,6,6.5,N/A,7,7,7,N/A,7,7,7,N/A,8,8,8,N/A,7,7,8,N/A,8,8,7,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Cody Linley,Julianne Hough,Actor/Actress,Texas,United States,18,7,Eliminated Week 9,4,7,6.5,7,N/A,7,7,7,N/A,7,7,7,N/A,7,8,8,N/A,10,9,9,N/A,8,8,7,N/A,8,7,7,N/A,8,8,8,N/A,8,7.5,7.5,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Misty May-Treanor,Maksim Chmerkoskiy,Athlete,California,United States,31,7,Withdrew,10,6.5,7.5,7,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Maurice Greene,Cheryl Burke,Athlete,Kansas,United States,34,7,Eliminated Week 8,5,6.5,6.5,6.5,N/A,7,6,6,N/A,8,8,8,N/A,6,7,7,N/A,9,9,9,N/A,7,7,7,N/A,8,9,8,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Warren Sapp,Kym Johnson,Athlete,Florida,United States,35,7,2nd Place,2,7,7,7.5,N/A,8,8,8,N/A,9,8,8,N/A,8,7,7,N/A,8,8,9,N/A,8,9,8,N/A,7,7,7,N/A,9.5,8.5,9,N/A,8.5,8,8,N/A,9,9.5,9,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jeffrey Ross,Edyta Sliwinska,Comedian,New Jersey,United States,43,7,Eliminated Week 1,13,4,4,4,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Toni Braxton,Alec Mazo,Singer/Rapper,Maryland,United States,40,7,Eliminated Week 5,8,7.5,7,8,N/A,7,8,8,N/A,8,7,7,N/A,7,7,8,N/A,7,7,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Lance Bass,Lacey Schwimmer,Singer/Rapper,Mississippi,United States,29,7,3rd Place,3,7.5,6,8,N/A,7,6,7,N/A,8,7,7,N/A,9,8,9,N/A,7,7,7,N/A,9,9,9,N/A,9,7,9,N/A,8.5,7.5,9,N/A,10,9,9.5,N/A,9,9,9.5,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kim Kardashian,Mark Ballas,TV Personality,California,United States,27,7,Eliminated Week 2,11,6,6.5,6,N/A,6,6,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Rocco DiSpirito,Karina Smirnoff,TV Personality,New York,United States,41,7,Eliminated Week 4,9,6,5.5,6,N/A,5,6,5,N/A,7,7,6,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Brooke Burke,Derek Hough,TV Personality,Connecticut,United States,37,7,1st Place,1,8,8,8.5,N/A,8,8,8,N/A,9,10,9,N/A,9,8,9,N/A,10,9,10,N/A,8,10,8,N/A,10,10,10,N/A,9.5,8.5,9.5,N/A,8,8.5,8,N/A,10,10,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Gilles Marini,Cheryl Burke,Actor/Actress,,France,33,8,2nd Place,2,8,8,8,N/A,9,9,9,N/A,9,9,9,N/A,10,10,10,N/A,10,9,10,N/A,9,8,9,N/A,9,9,9,N/A,9,9,9.5,N/A,9.5,9,9.5,N/A,10,10,10,N/A,9.5,10,9.5,N/A
|
||||||
|
Denise Richards,Maksim Chmerkoskiy,Actor/Actress,Illinois,United States,38,8,Eliminated Week 3,12,6,6,6,N/A,7,7,7,N/A,5,6,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
David Alan Grier,Kym Johnson,Actor/Actress,Michigan,United States,52,8,Eliminated Week 5,9,6,7,6,N/A,6,5,6,N/A,8,8,8,N/A,8,7,7,N/A,7,8,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Lawrence Taylor,Edyta Sliwinska,Athlete,Virginia,United States,50,8,Eliminated Week 7,7,6,5,5,N/A,7,6,7,N/A,7,6,7,N/A,7,5,7,N/A,6,7,7,N/A,7,7,8,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Ty Murray,Chelsie Hightower,Athlete,Arizona,United States,39,8,Eliminated Week 10,4,5,4,5,N/A,7,6,7,N/A,8,8,7,N/A,9,8,8,N/A,7,7,7,N/A,6,6,6,N/A,8,8,8,N/A,9,8,9,N/A,7.5,8,7.5,N/A,8,8,8,N/A,0,0,0,N/A
|
||||||
|
Shawn Johnson,Mark Ballas,Athlete,Iowa,United States,17,8,1st Place,1,8,8,7,N/A,8,8,8,N/A,9,9,9,N/A,8,8,9,N/A,9,8,9,N/A,8,9,9,N/A,9,9,10,N/A,9,8,9,N/A,9.5,9,9.5,N/A,9.5,9,9.5,N/A,10,10,10,N/A
|
||||||
|
Steve Wozniak,Karina Smirnoff,Entrepreneur,California,United States,58,8,Eliminated Week 4,10,5,4,4,N/A,6,5,6,N/A,4,3,3,N/A,4,4,4,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Belinda Carlisle,Jonathan Roberts,Singer/Rapper,California,United States,50,8,Eliminated Week 2,13,6,6,5,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Chuck Wicks,Julianne Hough,Singer/Rapper,Delaware,United States,29,8,Eliminated Week 8,6,6,7,7,N/A,6,7,7,N/A,8,7,8,N/A,8,7,7,N/A,7,8,8,N/A,8,7,8,N/A,9,9,9,N/A,8.5,8.5,8.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Lil' Kim,Derek Hough,Singer/Rapper,New York,United States,34,8,Eliminated Week 9,5,7,7,7,N/A,8,7,8,N/A,8,8,9,N/A,9,8,10,N/A,9,8,9,N/A,10,8,10,N/A,9,8,9,N/A,9,9,10,N/A,8.5,9,8.5,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Steve-O,Lacey Schwimmer,TV Personality,,England,34,8,Eliminated Week 6,8,6,5,6,N/A,5,4,5,N/A,5,5,5,N/A,5,5,5,N/A,6,6,6,N/A,7,4,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Holly Madison,Dmitry Chaplin,TV Personality,Oregon,United States,30,8,Eliminated Week 4,11,6,6,6,N/A,6,6,6,N/A,5,6,6,N/A,5,6,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Melissa Rycroft,Tony Dovolani,TV Personality,Texas,United States,26,8,3rd Place,3,8,7,8,N/A,9,8,9,N/A,9,9,9,N/A,10,9,10,N/A,8,8,9,N/A,9,9,9,N/A,10,9,10,N/A,7.5,7.5,8,N/A,9.5,9.5,9.5,N/A,9,9.5,9,N/A,9.5,9.5,9.5,N/A
|
||||||
|
Ashley Hamiliton,Edyta Sliwinska,Actor/Actress,California,United States,34,9,Eliminated Week 1,16,6.3333,7.3333,5.3333,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Debi Mazar,Maksim Chmerkoskiy,Actor/Actress,New York,United States,45,9,Eliminated Week 3,12,8,7,7,N/A,7,7,7,N/A,6,5,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Melissa Joan Hart,Mark Ballas,Actor/Actress,New York,United States,33,9,Eliminated Week 6,9,8,8,8,N/A,7,6,6,N/A,6,6,7,N/A,9,9,10,N/A,8,8,7,N/A,8.3333,8.3333,7.3333,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Mark Dacascos,Lacey Schwimmer,Actor/Actress,Hawaii,United States,45,9,Eliminated Week 7,6,9.6666,9.6666,9.6666,N/A,7,7,7,N/A,6,6,6,N/A,8,7,7,N/A,9,9,8,N/A,11,11,10,N/A,7,7.5,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Chuck Liddell,Anna Trebunskaya,Athlete,California,United States,39,9,Eliminated Week 4,11,8,7,7,N/A,6,7,6,N/A,6,5,6,N/A,6,5,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Natalie Coughlin,Alec Mazo,Athlete,California,United States,27,9,Eliminated Week 5,10,9.6666,8.6666,8.6666,N/A,7,7,7,N/A,9,8,9,N/A,8,8,8,N/A,7,8,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Louie Vito,Chelsie Hightower,Athlete,Ohio,United States,21,9,Eliminated Week 6,8,8.6666,9.6666,8.6666,N/A,6,7,6,N/A,8,5,7,N/A,5,5,6,N/A,7,8,7,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Michael Irvin,Anna Demidova,Athlete,Florida,United States,43,9,Eliminated Week 7,7,7,6,6,N/A,7,7,6,N/A,5,4,5,N/A,5,6,5,N/A,7,7,7,N/A,6.6666,8.6666,6.6666,N/A,8,8,7.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Joanna Krupa,Derek Hough,Model,,Poland,30,9,Eliminated Week 9,4,11.3333,11.3333,11.3333,N/A,6,7,7,N/A,7,8,8,N/A,9,8,9,N/A,8,8,8,N/A,11.3333,12.3333,12.3333,N/A,9,9,9.5,N/A,8.5,8.5,9,N/A,9,9,9,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kathy Ireland,Tony Dovolani,Model,California,United States,46,9,Eliminated Week 2,14,7.3333,6.3333,6.3333,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Tom DeLay,Cheryl Burke,Politician,Texas,United States,62,9,Withdrew,13,7.3333,6.3333,6.3333,N/A,6,6,6,N/A,6,4,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Macy Gray,Jonathan Roberts,Singer/Rapper,Ohio,United States,42,9,Eliminated Week 1,15,7.3333,5.3333,6.3333,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Aaron Carter,Karina Smirnoff,Singer/Rapper,Florida,United States,21,9,Eliminated Week 8,5,10.3333,11.3333,10.3333,N/A,9,9,9,N/A,8,6,7,N/A,6,6,6,N/A,8,8,8,N/A,10.6666,11.6666,10.6666,N/A,8.5,9,9,N/A,8,8.5,8.5,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Mya,Dmitry Chaplin,Singer/Rapper,Washington D.C.,United States,29,9,2nd Place,2,11.3333,8.3333,11.3333,N/A,9,9,9,N/A,10,7,10,N/A,10,8,10,N/A,9,9,9,N/A,11,10,12,N/A,8.5,7.5,8.5,N/A,9.5,10,10,N/A,9.3333,9.6666,10,N/A,9.4444,9.4444,9.4444,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Donny Osmond,Kym Johnson,Singer/Rapper,Utah,United States,51,9,1st Place,1,10.3333,9.3333,10.3333,N/A,8,9,8,N/A,7,7,7,N/A,8,8,8,N/A,10,9,10,N/A,10.3333,10.3333,10.3333,N/A,8.5,8.5,9,N/A,8.5,8,8.5,N/A,8,8.3333,8.3333,N/A,9.6666,9.6666,9.6666,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kelly Osbourne,Louis van Amstel,TV Personality,,England,24,9,3rd Place,3,9.6666,10.6666,10.6666,N/A,6,7,6,N/A,7,6,7,N/A,8,7,8,N/A,8,8,8,N/A,8.6666,7.6666,8.6666,N/A,8.5,8.5,9,N/A,8,8.5,9,N/A,8.6666,8.6666,8.6666,N/A,8.5,8.5,8.2222,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Aiden Turner,Edyta Sliwinska,Actor/Actress,,England,32,10,Eliminated Week 4,9,5,5,5,N/A,7,6,6,N/A,7,6,7,N/A,5.5,5.5,5.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Shannen Doherty,Mark Ballas,Actor/Actress,Tennessee,United States,38,10,Eliminated Week 2,11,6,6,6,N/A,7,6,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Niecy Nash,Louis van Amstel,Actor/Actress,California,United States,40,10,Eliminated Week 8,5,7,5,6,N/A,7,7,7,N/A,7,7,7,N/A,6,6,6,N/A,6,6,6,N/A,8.6666,8.6666,8.6666,N/A,8.5,8,8,N/A,7,7,7.5,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Buzz Aldrin,Ashly DelGrosso,Astronaut,New Jersey,United States,80,10,Eliminated Week 3,10,5,4,5,N/A,4,4,4,N/A,5,4,4,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Chad Ochocinco,Cheryl Burke,Athlete,Florida,United States,32,10,Eliminated Week 9,4,6,6,6,N/A,6,5,5,N/A,7,6,7,N/A,7.5,6.5,8,N/A,6,6,6,N/A,10.3333,10.3333,10.3333,N/A,8.5,9,8.5,N/A,7.5,7.5,7.5,N/A,8.5,8.5,9,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Evan Lysacek,Anna Trebunskaya,Athlete,Illinois,United States,24,10,2nd Place,2,8,7,8,N/A,8,8,8,N/A,9,8,9,N/A,9,8,9,N/A,9,9,9,N/A,9,9,9,N/A,9,9,9,N/A,9,8.5,9,N/A,10,9.5,10,N/A,9.3333,8.7777,8.7777,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Pamela Anderson,Damian Whitewood,Model,,Canada,42,10,Eliminated Week 7,6,7,6,8,N/A,7,7,8,N/A,7,7,7,N/A,7.5,7.5,8.5,N/A,7,6,8,N/A,9.6666,9.6666,10.6666,N/A,8.5,8.5,8.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Nicole Scherzinger,Derek Hough,Singer/Rapper,Hawaii,United States,31,10,1st Place,1,9,7,9,N/A,10,8,10,N/A,8,6,9,N/A,8.5,8,8.5,N/A,10,9,10,N/A,12.3333,10.3333,13.3333,N/A,9,9,9,N/A,10,9.5,10,N/A,10,9.5,10,N/A,9.5,9.5,9.7777,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Erin Andrews,Maksim Chmerkoskiy,Sports Broadcaster,Maine,United States,32,10,3rd Place,3,7,7,7,N/A,8,7,8,N/A,8,7,8,N/A,6.5,6.5,6.5,N/A,7,7,8,N/A,12,10,12,N/A,8.5,8.5,8.5,N/A,8.5,9,9,N/A,9.5,9,9.5,N/A,9.2222,8.8888,8.8888,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kate Gosselin,Tony Dovolani,TV Personality,Pennsylvania,United States,35,10,Eliminated Week 5,8,6,5,5,N/A,5,5,5,N/A,5,5,5,N/A,5,5.5,5.5,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jake Pavelka,Chelsie Hightower,TV Personality,Texas,United States,32,10,Eliminated Week 6,7,7,6,7,N/A,6,7,7,N/A,7,7,7,N/A,6.5,6.5,6,N/A,8,7,8,N/A,8.3333,8.3333,8.3333,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
David Hasselhoff,Kym Johnson,Actor/Actress,Maryland,United States,58,11,Eliminated Week 1,12,5,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Florence Henderson,Corky Ballas,Actor/Actress,Indiana,United States,76,11,Eliminated Week 5,8,6,6,6,N/A,7,6,6,N/A,7,6,7,N/A,6,6,5.5,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kyle Massey,Lacey Schwimmer,Actor/Actress,Georgia,United States,19,11,2nd Place,2,8,7,8,N/A,8,7,7,N/A,8,7,8,N/A,7,6.5,6.5,N/A,8,5,7,N/A,10.3333,9.3333,10.3333,N/A,9,8.5,8,8,9.5,9,9.5,N/A,10,9,10,N/A,9.3333,8.6666,9.3333,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jennifer Grey,Derek Hough,Actor/Actress,New York,United States,50,11,1st Place,1,8,8,8,N/A,8,8,8,N/A,8,8,8,N/A,9.5,9,9.5,N/A,8,8,9,N/A,9,10,10,N/A,9.5,9,9,9,9.5,9.5,9.5,N/A,10,10,10,N/A,10,10,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Rick Fox,Cheryl Burke,Athlete,,Canada,41,11,Eliminated Week 7,6,8,7,7,N/A,7,7,7,N/A,8,8,8,N/A,6,7,6.5,N/A,8,8,8,N/A,10,10,10,N/A,9,8.5,8.5,8.5,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kurt Warner,Anna Trebunskaya,Athlete,Iowa,United States,39,11,Eliminated Week 8,5,7,5,7,N/A,7,7,7,N/A,8,8,7,N/A,6,5.5,5.5,N/A,8,8,8,N/A,7.3333,7.3333,7.3333,N/A,9.5,8.5,8.5,8.5,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Margaret Cho,Louis van Amstel,Comedian,California,United States,41,11,Eliminated Week 3,10,5,5,5,N/A,6,6,6,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Michael Bolton,Chelsie Hightower,Singer/Rapper,Connecticut,United States,57,11,Eliminated Week 2,11,6,5,5,N/A,4,5,3,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Brandy,Maksim Chmerkoskiy,Singer/Rapper,Mississippi,United States,31,11,Eliminated Week 9,4,7,8,8,N/A,7,7,7,N/A,8,8,8,N/A,8,8,8,N/A,9,9,9,N/A,11.3333,12.3333,12.3333,N/A,9,9,9.5,9,9,9.5,10,N/A,9.5,9.5,9.5,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
The Situation,Karina Smirnoff,TV Personality,New York,United States,28,11,Eliminated Week 4,9,5,5,5,N/A,6,6,6,N/A,7,6,7,N/A,5,4.5,4.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Audrina Patridge,Tony Dovolani,TV Personality,California,United States,25,11,Eliminated Week 6,7,6,7,6,N/A,8,8,7,N/A,8,9,9,N/A,8,7.5,7.5,N/A,7,8,8,N/A,10.6666,10.6666,10.6666,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Bristol Palin,Mark Ballas,TV Personality,Alaska,United States,19,11,3rd Place,3,6,6,6,N/A,7,8,7,N/A,6,6,7,N/A,5,5.5,5.5,N/A,6,6,6,N/A,9.6666,8.6666,9.6666,N/A,8.5,8,8,8,7.5,8,8,N/A,8.5,9,9,N/A,8.3333,9,8.3333,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Ralph Macchio,Karina Smirnoff,Actor/Actress,New York,United States,49,12,Eliminated Week 9,4,8,8,8,N/A,7,7,7,N/A,7,7,7,N/A,8,8,9,N/A,8,7,7,N/A,8,8,8,N/A,9,8.5,7.5,8,7.5,7.5,8,N/A,8,8,8,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Chelsea Kane,Mark Ballas,Actor/Actress,Arizona,United States,22,12,3rd Place,3,7,7,7,N/A,6,5,7,N/A,7,8,8,N/A,9,8,9,N/A,9,8,9,N/A,10,9,9,N/A,8,8.5,7.5,8,9,9,9.5,N/A,12,12,12.5,N/A,10,9.6666,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kirstie Alley,Maksim Chmerkoskiy,Actor/Actress,Kansas,United States,60,12,2nd Place,2,8,7,8,N/A,7,6,7,N/A,7,7,7,N/A,7,7,8,N/A,8,7,8,N/A,8,9,9,N/A,8,7,7,8,8.5,9,9,N/A,9,9,9,N/A,9.3333,9.3333,9.3333,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Hines Ward,Kym Johnson,Athlete,,South Korea,35,12,1st Place,1,7,7,7,N/A,8,7,8,N/A,9,8,8,N/A,9,8,8,N/A,9,9,9,N/A,9,9,9,N/A,8,8,8,9,9,9,9,N/A,10,10,10,N/A,10,9.6666,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Sugar Ray Leonard,Anna Trebunskaya,Athlete,North Carolina,United States,54,12,Eliminated Week 4,9,6,5,6,N/A,6,5,6,N/A,7,6,7,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Chris Jericho,Cheryl Burke,Athlete,New York,United States,40,12,Eliminated Week 6,7,7,6,6,N/A,8,7,8,N/A,7,7,7,N/A,8,7,8,N/A,9,8,9,N/A,7,8,7,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Petra Nemcova,Dmitry Chaplin,Model,,Czechoslovakia,31,12,Eliminated Week 5,8,6,6,6,N/A,6,6,6,N/A,8,9,8,N/A,8,7,8,N/A,7,7,8,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Mike Catherwood,Lacey Schwimmer,Radio Personality,California,United States,32,12,Eliminated Week 2,11,5,4,4,N/A,6,5,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Romeo,Chelsie Hightower,Singer/Rapper,Louisiana,United States,21,12,Eliminated Week 8,5,7,6,6,N/A,7,8,8,N/A,7,6,7,N/A,7,8,8,N/A,9,8,9,N/A,10,9,9,N/A,7.5,8,7,7.5,8.5,9,8.5,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Wendy Williams,Tony Dovolani,TV Personality,New Jersey,United States,46,12,Eliminated Week 3,10,5,4,5,N/A,6,5,6,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kendra Wilkinson,Louis van Amstel,TV Personality,California,United States,25,12,Eliminated Week 7,6,6,6,6,N/A,7,6,6,N/A,8,7,8,N/A,6,6,6,N/A,8,7,7,N/A,8,8,9,N/A,7.5,7.5,7.5,8,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Elisabetta Canalis,Valentin Chmerkovskiy,Actor/Actress,,Italy,33,13,Eliminated Week 2,11,5,5,5,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
David Arquette,Kym Johnson,Actor/Actress,Virginia,United States,40,13,Eliminated Week 7,6,6,6,6,N/A,6,6,6,N/A,8,8,8,N/A,8,7,8,N/A,8,9,8,N/A,8,7,8,N/A,8,7.5,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Ricki Lake,Derek Hough,Actor/Actress,New York,United States,42,13,3rd Place,3,7,6,7,N/A,8,7,8,N/A,9,9,9,N/A,10,9,10,N/A,8,8,8,N/A,10,9,10,N/A,9,8.5,9,N/A,8.5,8.5,9,N/A,10.8333,11.3333,11.3333,N/A,9.3333,9.3333,9.3333,N/A,N/A,N/A,N/A,N/A
|
||||||
|
J.R. Martinez,Karina Smirnoff,Actor/Actress,Louisiana,United States,28,13,1st Place,1,8,7,7,N/A,7,7,8,N/A,9,8,9,N/A,8,9,9,N/A,9,9,10,N/A,10,9,10,N/A,8.5,7.5,8,N/A,10,10,10,N/A,9.5,9,9.5,N/A,9.3333,9.1666,9.6666,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Metta World Peace (Ron),Peta Murgatroyd,Athlete,New York,United States,31,13,Eliminated Week 1,12,5,4,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Hope Solo,Maksim Chmerkoskiy,Athlete,Washington,United States,30,13,Eliminated Week 9,4,7,7,7,N/A,6,7,6,N/A,8,8,8,N/A,8,8,8,N/A,8,8,8,N/A,7,6,7,N/A,8.5,8,8.5,N/A,8.5,9,8.5,N/A,8.1666,8.1666,8.1666,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Chynna Phillips,Tony Dovolani,Singer/Rapper,California,United States,43,13,Eliminated Week 4,9,8,7,7,N/A,7,7,7,N/A,8,9,9,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kristin Cavallari,Mark Ballas,TV Personality,Colorado,United States,24,13,Eliminated Week 3,10,7,6,6,N/A,8,7,7,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Carson Kressley,Anna Trebunskaya,TV Personality,Pennsylvania,United States,41,13,Eliminated Week 5,8,6,5,6,N/A,6,6,6,N/A,8,7,8,N/A,7,6,7,N/A,6,6,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Chaz Bono,Lacey Schwimmer,TV Personality,California,United States,42,13,Eliminated Week 6,7,6,5,6,N/A,6,5,6,N/A,6,6,6,N/A,7,7,7,N/A,7,7,7,N/A,7,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Nancy Grace,Tristan MacManus,TV Personality,Georgia,United States,51,13,Eliminated Week 8,5,5,5,6,N/A,6,8,7,N/A,7,7,7,N/A,7,7,7,N/A,7,7,8,N/A,9,7,8,N/A,7.5,7,7.5,N/A,7.5,7,7.5,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Rob Kardashian,Cheryl Burke,TV Personality,California,United States,24,13,2nd Place,2,6,5,5,N/A,7,7,7,N/A,8,8,8,N/A,8,8,8,N/A,9,8,8,N/A,8,7,7,N/A,9,8,8.5,N/A,8.5,8.5,8.5,N/A,11.1666,10.6666,10.6666,N/A,9.4444,9.4444,9.4444,N/A,N/A,N/A,N/A,N/A
|
||||||
|
William Levy,Cheryl Burke,Actor/Actress,,Cuba,31,14,3rd Place,3,8,8,8,N/A,9,7,9,N/A,9,9,10,N/A,7,7,8,N/A,10,9,10,N/A,12,11,13,N/A,9,8.5,9,N/A,9.5,9.5,9.5,N/A,9.5,9.5,10,N/A,10,9.6666,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jack Wagner,Anna Trebunskaya,Actor/Actress,Missouri,United States,52,14,Eliminated Week 3,11,8,7,8,N/A,7,7,7,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Sherri Sheperd,Valentin Chmerkovskiy,Actor/Actress,Illinois,United States,44,14,Eliminated Week 4,10,8,7,8,N/A,8,7,8,N/A,8,8,8,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jaleel White,Kym Johnson,Actor/Actress,California,United States,35,14,Eliminated Week 7,7,9,8,9,N/A,7,7,8,N/A,9,8,8,N/A,8,7,7,N/A,8,8,8,N/A,12.6666,11.6666,12.6666,N/A,9,8,8.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Roshon Fegan,Chelsie Hightower,Actor/Actress,California,United States,20,14,Eliminated Week 8,6,8,7,8,N/A,9,8,9,N/A,8,8,9,N/A,9,8,9,N/A,9,8,9,N/A,8.6666,9.6666,9.6666,N/A,9.5,8,8.5,N/A,9.5,9,9.5,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Melissa Gilbert,Maksim Chmerkoskiy,Actor/Actress,California,United States,47,14,Eliminated Week 8,5,7,6,7,N/A,7,6,7,N/A,8,8,8,N/A,7,8,7,N/A,7,7,7,N/A,10,10,10,N/A,8,7.5,8,N/A,8.5,8.5,8.5,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Martina Navratilova,Tony Dovolani,Athlete,,Czechoslovakia,55,14,Eliminated Week 2,12,7,6,7,N/A,6,5,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Donald Driver,Peta Murgatroyd,Athlete,Texas,United States,37,14,1st Place,1,7,7,7,N/A,8,8,8,N/A,9,8,9,N/A,9,9,9,N/A,10,8,9,N/A,11.3333,11.3333,11.3333,N/A,9,8.5,9,N/A,9.5,9,9,N/A,9.5,9,10,N/A,10,9.6666,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Gavin DeGraw,Karina Smirnoff,Singer/Rapper,New York,United States,35,14,Eliminated Week 5,9,7,6,7,N/A,7,7,7,N/A,8,8,8,N/A,8,8,7,N/A,6,6,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Gladys Knight,Tristan MacManus,Singer/Rapper,Georgia,United States,67,14,Eliminated Week 6,8,8,7,8,N/A,7,5,7,N/A,8,8,8,N/A,7,6,7,N/A,7,7,8,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Katherine Jenkins,Mark Ballas,Singer/Rapper,,Wales,31,14,2nd Place,2,9,8,9,N/A,9,8,9,N/A,10,9,10,N/A,8,8,8,N/A,10,9,10,N/A,13.3333,12.3333,13.3333,N/A,9.5,8.5,9,N/A,9,9,9.5,N/A,9.5,9,9.5,N/A,10,10,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Maria Menounos,Derek Hough,TV Personality,Massachusetts,United States,33,14,Eliminated Week 9,4,7,7,7,N/A,8,8,9,N/A,9,9,9,N/A,9,8,9,N/A,9,9,9,N/A,9.3333,10.3333,10.3333,N/A,10,9,9.5,N/A,9.5,7.5,9.5,N/A,10,9.5,10,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Gilles Marini,Peta Murgatroyd,Actor/Actress,,France,36,15,Eliminated Week 8,6,8,8,8,N/A,8.5,8.5,8.5,N/A,8.5,8.5,8.5,N/A,10,9.5,10,10,9.5,9.25,9.5,N/A,9,9,9.5,N/A,11.1666,11.1666,11.1666,N/A,9.5,10,9.7777,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Sabrina Bryan,Louis van Amstel,Actor/Actress,California,United States,28,15,Eliminated Week 6,8,7.5,7.5,7.5,N/A,9,8.5,8.5,N/A,8.5,8.5,8.5,N/A,9,9,8.5,9,9.75,9.75,9.75,N/A,10,10,10,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kirstie Alley,Maksim Chmerkoskiy,Actor/Actress,Kansas,United States,61,15,Eliminated Week 8,7,6.5,6,6.5,N/A,7,7,7,N/A,8,8,8,N/A,7.5,7.5,7.5,7.5,8.75,8.75,8.75,N/A,9.5,8.5,9.5,N/A,9.3333,9.3333,9.3333,N/A,8.5,8.5,8.5,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kelly Monaco,Valentin Chmerkovskiy,Actor/Actress,Pennsylvania,United States,36,15,3rd Place,3,7,7,7.5,N/A,7.5,7,7.5,N/A,9,9,9,N/A,9,9.5,9.5,9.5,8.5,8.5,8.75,N/A,9,9,9,N/A,12,12,12,N/A,9.5,9.3333,9.5,N/A,8.7777,9.3333,9,N/A,9.6666,9.6666,9.7777,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Apolo Anton Ohno,Karina Smirnoff,Athlete,Washington,United States,30,15,Eliminated Week 9,5,7.5,7,7.5,N/A,8.5,8,8,N/A,9,8,8.5,N/A,8.5,9,8.5,8.5,9,9.75,9.5,N/A,10,10,10,N/A,11,11,11,N/A,9.7777,9.5,10,N/A,9.3333,9.5,9.7777,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Emmitt Smith,Cheryl Burke,Athlete,Florida,United States,43,15,Eliminated Week 9,4,8,8.5,8,N/A,7.5,7.5,7.5,N/A,8.5,8,8.5,N/A,9,9,9,9,9.25,9.25,9.5,N/A,8.5,8.5,9.5,N/A,10.8333,11.8333,11.8333,N/A,9.5,9.7777,9.7777,N/A,9,9,9,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Shawn Johnson,Derek Hough,Athlete,Iowa,United States,20,15,2nd Place,2,8,6.5,7.5,N/A,8.5,8,8.5,N/A,9,8,9.5,N/A,10,9.5,10,10,9.25,9,10,N/A,9.5,8.5,10,N/A,12.6666,12.6666,12.6666,N/A,10,8.3333,9.5,N/A,9.5,10,10,N/A,9.6666,9.5,9.7777,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Pamela Anderson,Tristan MacManus,Model,,Canada,45,15,Eliminated Week 1,13,5.5,5.5,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Helio Castroneves,Chelsie Hightower,Racing Driver,,Brazil,37,15,Eliminated Week 3,10,7,7.5,7,N/A,8,7.5,7.5,N/A,8.5,8.5,8.5,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Joey Fatone,Kym Johnson,Singer/Rapper,New York,United States,35,15,Eliminated Week 2,12,6.5,7,7,N/A,7.5,7.5,7.5,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
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|
Drew Lachey,Anna Trebunskaya,Singer/Rapper,Ohio,United States,36,15,Eliminated Week 3,11,7,7,7.5,N/A,7.5,7.5,7.5,N/A,8,8,8,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Bristol Palin,Mark Ballas,TV Personality,Alaska,United States,21,15,Eliminated Week 4,9,6.5,6.5,6.5,N/A,6,6,6,N/A,7.5,7.5,7.5,N/A,8,8,8,8,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Melissa Rycroft,Tony Dovolani,TV Personality,Texas,United States,29,15,1st Place,1,7,7,7,N/A,8,8,7.5,N/A,9,9,9,N/A,9,9,9.5,9.5,9.25,9.5,9.5,N/A,9.5,10,10,N/A,13.3333,12.8333,12.8333,N/A,10,10,10,N/A,9.5,9.7777,9.5,N/A,9.7777,9.7777,9.7777,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Ingo Rademacher,Kym Johnson,Actor/Actress,,Germany,41,16,Eliminated Week 9,5,7,6,7,N/A,6,7,7,N/A,7,7,7,N/A,8,7,8,N/A,7,7,7,N/A,7.5,8,7.5,N/A,8,9,8,N/A,8,8,8,N/A,8.5,8.5,8.5,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Zendaya,Valentin Chmerkovskiy,Actor/Actress,California,United States,16,16,2nd Place,2,8,8,8,N/A,9,8,9,N/A,8.6666,8.6666,8.6666,N/A,9,8,9,N/A,10,9,10,N/A,8.5,8.5,8.5,N/A,10,10,10,N/A,9.5,10,9.5,N/A,9,9,9.5,N/A,10,10,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Dorothy Hamill,Tristan MacManus,Athlete,Illinois,United States,56,16,Withdrew,12,7,7,7,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Victor Ortiz,Lindsay Arnold,Athlete,Kansas,United States,26,16,Eliminated Week 6,8,6,6,6,N/A,6,6,6,N/A,8,7,8,N/A,6,6,6,N/A,7,7,7,N/A,6.5,7,6.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Alexandra Raisman,Mark Ballas,Athlete,Massachusetts,United States,18,16,Eliminated Week 10,4,7,7,7,N/A,8,8,8,N/A,7,8,8,N/A,9,9,9,N/A,8,8,9,N/A,8.5,9,8.5,N/A,11,10,11,N/A,9,9.5,9.5,N/A,10,9.5,10,N/A,9.5,9.5,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jacoby Jones,Karina Smirnoff,Athlete,Louisiana,United States,28,16,3rd Place,3,7,6,7,N/A,8,7,8,N/A,8.6666,8.6666,8.6666,N/A,8,8,8,N/A,9,8,9,N/A,7.5,7.5,7.5,N/A,9,9,9,N/A,8.5,9,8.5,N/A,10,9.5,10,N/A,9.3333,9.3333,9.3333,N/A,N/A,N/A,N/A,N/A
|
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|
D. L. Hughley,Cheryl Burke,Comedian,Virginia,United States,50,16,Eliminated Week 5,9,4,4,4,N/A,5,5,6,N/A,6,5,5,N/A,7,7,7,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Andy Dick,Sharna Burgess,Comedian,South Carolina,United States,47,16,Eliminated Week 7,7,6,5,6,N/A,7,6,7,N/A,6,6,6,N/A,7,7,7,N/A,6,6,6,N/A,7,7.5,7,N/A,5,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Wynonna Judd,Tony Dovolani,Singer/Rapper,Kentucky,United States,48,16,Eliminated Week 3,11,6,6,6,N/A,6,6,6,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kellie Pickler,Derek Hough,Singer/Rapper,North Carolina,United States,26,16,1st Place,1,7,7,7,N/A,9,8,9,N/A,8,9,8,N/A,9,8,9,N/A,9,9,9,N/A,8.5,9.5,9,N/A,9,10,10,N/A,9.5,8,10,N/A,9.5,10,9.5,N/A,10,10,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Lisa Vanderpump,Gleb Savchenko,TV Personality,,England,52,16,Eliminated Week 4,10,6,6,6,N/A,6,6,6,N/A,7,7,7,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Sean Lowe,Peta Murgatroyd,TV Personality,Texas,United States,29,16,Eliminated Week 8,6,7,6,6,N/A,7,6,7,N/A,7,7,7,N/A,6,7,7,N/A,8,8,8,N/A,7.5,8,7.5,N/A,8,8,8,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Valerie Harper,Tristan MacManus,Actor/Actress,New York,United States,74,17,Eliminated Week 4,10,7,7,7,N/A,6,6,7,N/A,6,5,5,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A
|
||||||
|
Brant Daugherty,Peta Murgatroyd,Actor/Actress,Ohio,United States,28,17,Eliminated Week 8,7,7,8,7,N/A,8,7,8,N/A,9,9,9,N/A,7,7,7,N/A,9,9,9,N/A,10,11,10,N/A,9.5,9.5,9.5,N/A,9,9,9,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A
|
||||||
|
Elizabeth Berkley Lauren,Valentin Chmerkovskiy,Actor/Actress,Michigan,United States,39,17,Eliminated Week 9,6,8,8,8,N/A,8,9,8,N/A,8,9,8,N/A,9,9,9,N/A,8,9,9,N/A,10.6666,10.6666,10.6666,N/A,9,9,9,N/A,8,9,8,N/A,9.5,9,9.5,N/A,0,0,0,0,0,0,0,N/A
|
||||||
|
Leah Remini,Tony Dovolani,Actor/Actress,New York,United States,43,17,Eliminated Week 10,5,7,7,7,N/A,8,8,8,N/A,8,8,8,N/A,8,8,8,N/A,7,7,8,N/A,9.3333,9.3333,9.3333,N/A,9,8.5,9,N/A,9,10,9,N/A,9,9,9,N/A,8,8,8,8.5,0,0,0,N/A
|
||||||
|
Corbin Bleu,Karina Smirnoff,Actor/Actress,New York,United States,24,17,2nd Place,2,8,8,8,N/A,9,8,9,N/A,9,8,9,N/A,9,9,9,N/A,9,9,10,N/A,9.3333,8.3333,9.3333,N/A,10,9.5,10,N/A,10,10,10,N/A,9.5,9.5,10,N/A,9.5,9,9.5,9.5,9.8888,9.8888,9.8888,N/A
|
||||||
|
Amber Riley,Derek Hough,Actor/Actress,California,United States,27,17,1st Place,1,9,9,9,N/A,8,8,8,N/A,8,8,8,N/A,9,9,9,N/A,9,7,10,N/A,11.3333,9.3333,11.3333,N/A,10,10,9.5,N/A,9,9,10,N/A,8.5,8.5,8.5,N/A,10,10,9.5,10,10.4444,10.4444,10.4444,N/A
|
||||||
|
Keyshawn Johnson,Sharna Burgess,Athlete,California,United States,41,17,Eliminated Week 2,12,6,5,6,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A
|
||||||
|
Bill Engvall,Emma Slater,Comedian,Texas,United States,56,17,Eliminated Week 11,4,6,6,6,N/A,7,7,7,N/A,8,8,8,N/A,7,7,7,N/A,8,8,8,N/A,8.3333,8.3333,7.3333,N/A,8.5,8,8.5,N/A,8,8,8,N/A,7,7,7,N/A,7.5,7.5,7.5,7.5,8.3333,8.8333,8.3333,N/A
|
||||||
|
Christina Milian,Mark Ballas,Singer/Rapper,New Jersey,United States,31,17,Eliminated Week 5,9,7,7,8,N/A,9,8,8,N/A,9,8,9,N/A,8,8,8,N/A,9,10,9,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A
|
||||||
|
"Nicole ""Snooki"" Polizzi",Sasha Farber,TV Personality,,Chile,25,17,Eliminated Week 7,8,8,8,7,N/A,6,7,7,N/A,9,8,8,N/A,8,8,8,N/A,9,9,9,N/A,10,10,10,N/A,9,9,9,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A
|
||||||
|
Jack Osbourne,Cheryl Burke,TV Personality,,England,27,17,3rd Place,3,8,8,7,N/A,8,8,8,N/A,7,7,8,N/A,8,8,8,N/A,9,9,9,N/A,8.6666,9.6666,8.6666,N/A,10,9,9.5,N/A,10,10,10,N/A,9,9,9,N/A,9,8.5,9,9,9.3333,9.3333,9.3333,N/A
|
||||||
|
Bill Nye,Tyne Stecklein,TV Personality,Washington D.C.,United States,57,17,Eliminated Week 3,11,5,4,5,N/A,6,5,6,N/A,6,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A
|
||||||
|
Billy Dee Williams,Emma Slater,Actor/Actress,New York,United States,76,18,Withdrew,10,5,5,5,N/A,5,5,5,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Danica McKellar,Valentin Chmerkovskiy,Actor/Actress,California,United States,39,18,Eliminated Week 8,6,8,8,8,N/A,8,8,8,N/A,9,9,9,9,8,8,8,8,10,9,10,10,9,9,9,9,9,9.5,8.5,9,10,9,9,10,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
James Maslow,Peta Murgatroyd,Actor/Actress,New York,United States,23,18,Eliminated Week 10,4,7,7,7,N/A,9,8,8,N/A,9,9,9,9,9,8,9,9,10,10,10,10,9,9,8,9,8.5,9.5,8,9,8,9,10,9,9.5,9.5,9.5,9.5,9.5,9.5,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Candace Cameron Bure,Mark Ballas,Actor/Actress,California,United States,37,18,3rd Place,3,9,8,8,N/A,7,7,7,N/A,8,8,8,8,7,7,7,7,8,9,9,9,8,8,8,8,9,9.5,9,9.5,9,9,9,9,8.5,9.5,9.5,8.5,8.6666,8.6666,8.6666,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Sean Avery,Karina Smirnoff,Athlete,,Canada,33,18,Eliminated Week 2,11,7,6,7,N/A,7,7,7,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Diana Nyad,Henry Byalikov,Athlete,New York,United States,64,18,Eliminated Week 2,12,6,6,6,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Charlie White,Sharna Burgess,Athlete,Michigan,United States,26,18,Eliminated Week 9,5,9,9,9,N/A,9,7,9,N/A,9,9,9,9,7,8,9,9,9,10,9,9,9,9,9,9,8.5,10,8,9,10,10,10,10,9.5,9.5,9.5,9.5,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Amy Purdy,Derek Hough,Athlete,Nevada,United States,34,18,2nd Place,2,8,8,8,N/A,8,8,8,N/A,9,9,9,9,9,8,8,9,9,9,10,9,9,10,9,10,9.5,9.5,9,9.5,10,10,10,10,9.5,9.5,10,10,10,9.6666,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Meryl Davis,Maksim Chmerkoskiy,Athlete,Michigan,United States,27,18,1st Place,1,8,8,8,N/A,8,9,8,N/A,10,9,10,10,10,9,10,10,9,9,9,9,10,10,10,10,10,10,9,10,9,9,8,10,10,10,10,10,10,10,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Drew Carey,Cheryl Burke,Comedian,Ohio,United States,55,18,Eliminated Week 6,8,7,7,7,N/A,7,7,7,N/A,7,7,8,8,8,8,9,8,7,7,7,7,8,9,7,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Cody Simpson,Witney Carson,Singer/Rapper,,Australia,17,18,Eliminated Week 5,9,7,7,8,N/A,7,7,8,N/A,9,8,9,9,8,7,8,8,9,8,8,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
NeNe Leakes,Tony Dovolani,TV Personality,New York,United States,46,18,Eliminated Week 7,7,7,7,7,N/A,7,7,7,N/A,8,7,8,8,8,8,8,8,9,9,9,9,8,9,8,8,8,9,7.5,8.5,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Antonio Sabato Jr.,Cheryl Burke,Actor/Actress,,Italy,42,19,Eliminated Week 7,8,6,6,6,7,8,7,8,8,7,7,7,8,7,7,7,8,8,6,7,7,7,7,7,7,7,7.5,7.5,7.5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
|
||||||
|
Jonathan Bennett,Allison Holker,Actor/Actress,Ohio,United States,33,19,Eliminated Week 6,9,8,7,7,8,7,7,8,8,8,8,8,8,6,6,6,6,6,6,6,6,8,8,8,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
|
||||||
|
Lea Thompson,Artem Chigvintsev,Actor/Actress,Minnesota,United States,53,19,Eliminated Week 9,6,8,8,8,8,9,8,9,9,7,8,8,8,10,10,10,9,9,8,8,9,8,8,8,8,8.5,8.5,9,9,8,8,8,8,8.5,9,8.5,9,0,0,0,0,0,0,0,0
|
||||||
|
Janel Parrish,Valentin Chmerkovskiy,Actor/Actress,Hawaii,United States,25,19,3rd Place,3,7,7,7,8,9,8,8,9,10,10,10,10,9,9,9,9,8,8,8,9,8,7,9,9,8.5,8,8.5,8.5,10.75,10.75,10.75,10.75,9.5,9.5,10,9.5,9.5,9.5,10,10,9.6666,10,9.6666,9.6666
|
||||||
|
Alfonso Ribeiro,Witney Carson,Actor/Actress,New York,United States,42,19,1st Place,1,9,9,9,9,8,8,8,8,8,8,8,8,10,10,10,10,8,9,9,8,10,10,9,10,8.5,8.5,8.5,8.5,10.75,9.75,9.75,10.75,9.5,9.5,9.5,10,9.5,9,9.5,9.5,10,10,10,10
|
||||||
|
Lolo Jones,Keo Motsepe,Athlete,Iowa,United States,32,19,Eliminated Week 1,13,6,6,5,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
|
||||||
|
Randy Couture,Karina Smirnoff,Athlete,Washington,United States,51,19,Eliminated Week 3,11,8,7,8,8,7,7,7,7,7,7,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
|
||||||
|
Tommy Chong,Peta Murgatroyd,Comedian,,Canada,76,19,Eliminated Week 10,5,7,6,7,7,7,7,7,7,8,10,8,8,7,7,7,7,6,6,5,6,7,7,7,7,7.5,7.5,7.5,7.5,6.75,7.75,7.75,6.75,7,7.5,7,7,8,7.5,8,7.5,0,0,0,0
|
||||||
|
Betsey Johnson,Tony Dovolani,Fashion Designer,Connecticut,United States,72,19,Eliminated Week 4,10,5,5,5,5,7,7,7,7,7,9,6,7,8,7,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
|
||||||
|
Michael Waltrip,Emma Slater,Racing Driver,Kentucky,United States,51,19,Eliminated Week 8,7,7,6,6,6,6,6,6,6,7,7,7,7,6,6,6,7,5,5,5,5,8,7,8,7,7,7,7,7,6,7,6,6,0,0,0,0,0,0,0,0,0,0,0,0
|
||||||
|
Tavis Smiley,Sharna Burgess,Radio Personality,Mississippi,United States,50,19,Eliminated Week 2,12,7,7,8,7,7,7,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
|
||||||
|
Bethany Mota,Derek Hough,Social Media Personality,California,United States,18,19,Eliminated Week 11,4,8,8,8,8,8,9,8,8,10,10,10,10,8,8,8,9,8,8,8,8,9,9,9,9,9.5,9,9.5,9.5,9.75,9.75,9.75,10.75,9,9.5,9.5,9,9.5,9.5,9.5,9.5,9.5,9.5,9.5,9.5
|
||||||
|
Sadie Robertson,Mark Ballas,TV Personality,Louisiana,United States,17,19,2nd Place,2,8,8,9,9,8,7,8,8,8,8,8,8,9,10,9,9,9,9,9,9,9,10,8,8,7.5,7.5,8,8,9,9,10,10,9,9.5,9,9,9,9.5,9.5,9,10,9.6666,10,9.6666
|
||||||
|
Suzanne Somers,Tony Dovolani,Actor/Actress,California,United States,68,20,Eliminated Week 5,9,6,6,6,7,7,7,7,7,6,6,6,7,7,7,7,7,7,7,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Willow Shields,Mark Ballas,Actor/Actress,New Mexico,United States,14,20,Eliminated Week 7,7,6,6,6,7,8,8,8,8,8,8,8,8,10,9,10,10,8,8,9,9,9.5,8.5,9,9.5,9.5,9.5,9.5,10.5,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Riker Lynch,Allison Holker,Actor/Actress,Colorado,United States,23,20,2nd Place,2,8,7,8,8,8,8,8,8,9,7,9,9,8,8,9,9,10,9,9,10,10,8.5,9.5,10,9,10,9,9,10,9,10,10,10,10,10,10,10,10,10,10,N/A,N/A,N/A,N/A
|
||||||
|
Rumer Willis,Valentin Chmerkovskiy,Actor/Actress,Kentucky,United States,26,20,1st Place,1,8,8,8,8,8,8,8,8,8,9,8,8,9,8,9,9,10,9,10,10,9,8.5,8.5,9.5,8.5,9.5,9.5,9.5,10,10,10,10,10,9.5,9.5,10,10,10,10,10,N/A,N/A,N/A,N/A
|
||||||
|
Nastia Liukin,Derek Hough,Athlete,,Russia,25,20,Eliminated Week 9,4,7,7,8,8,9,8,8,9,9,8,9,8,9,8,9,10,9,9,10,10,9.5,8.5,9,9.5,10.75,9.75,9.75,10.75,9.5,9.5,9.5,9.5,10,10,10,10,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Michael Sam,Peta Murgatroyd,Athlete,Texas,United States,25,20,Eliminated Week 4,10,6,6,7,7,7,7,7,7,6,6,6,6,7,7,8,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Robert Herjavec,Kym Johnson,Entrepreneur,,Croatia,52,20,Eliminated Week 8,6,7,7,7,7,7,7,7,7,7,7,8,7,8,9,8,9,6,6,6,6,8.5,8,8.5,8.5,7,8,8,8,7.5,7.5,8.5,7.5,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Charlotte McKinney,Keo Motsepe,Model,Florida,United States,21,20,Eliminated Week 3,11,6,5,5,6,7,6,7,6,6,5,5,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Noah Galloway,Sharna Burgess,Motivational Speaker,Alabama,United States,33,20,3rd Place,3,7,6,6,7,7,6,7,7,7,7,8,8,8,8,8,8,7,7,7,7,8.5,8,8.5,9,10.5,8.5,9.5,9.5,8,7.5,8,8,9.5,9.5,9.5,9.5,9,9,9,9,N/A,N/A,N/A,N/A
|
||||||
|
Redfoo,Emma Slater,Singer/Rapper,California,United States,39,20,Eliminated Week 2,12,6,5,5,6,8,7,8,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Patti LaBelle,Artem Chigvintsev,Singer/Rapper,Pennsylvania,United States,70,20,Eliminated Week 6,8,7,6,6,6,7,7,7,7,6,5,5,6,8,7,7,8,7,6,7,7,9,8,8.5,8.5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Chris Soules,Witney Carson,TV Personality,Iowa,United States,33,20,Eliminated Week 8,5,7,6,6,7,5,6,5,5,7,7,7,7,7,6,7,7,7,6,7,7,9,8,9,9,8,7,8,8,8,7.5,8.5,8,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Gary Busey,Anna Trebunskaya,Actor/Actress,Texas,United States,71,21,Eliminated Week 4,10,5,5,5,N/A,5.5,5.5,5.5,N/A,6,7,6,6,5,6,5,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Alexa PenaVega,Mark Ballas,Actor/Actress,Florida,United States,27,21,Eliminated Week 9,6,7,7,8,N/A,7.5,7.5,8,N/A,9,9,9,9,7,7,7,N/A,10,10,10,10,8,7,7,8,9,9.5,9,N/A,9.6666,8.6666,8.6666,N/A,10,10,10,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Carlos PenaVega,Witney Carson,Actor/Actress,Missouri,United States,26,21,Eliminated Week 11,4,8,8,7,N/A,7.5,7.5,7.5,N/A,7,8,8,8,9,8,8,N/A,10,10,9,10,10,9,9,10,9,10,9,N/A,9.6666,9.6666,9.6666,N/A,9,9,9,N/A,10,10.5,11,N/A,10,10,10,N/A
|
||||||
|
Victor Espinoza,Karina Smirnoff,Athlete,,Mexico,43,21,Eliminated Week 2,12,5,5,5,N/A,6.5,5.5,6.5,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Alek Skarlatos,Lindsay Arnold,Military,California,United States,22,21,3rd Place,3,8,7,7,N/A,7.5,7,8,N/A,8,8,9,8,8,8,8,N/A,8,7,7,7,8,8,7,7,9,9,8.5,N/A,9,8,8,N/A,8.5,8.5,8.5,N/A,9.5,9.5,9.5,N/A,9.3333,9.3333,9.3333,N/A
|
||||||
|
Chaka Khan,Keo Motsepe,Singer/Rapper,Illinois,United States,62,21,Eliminated Week 2,13,5,4,4,N/A,5,5,5,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Andy Grammer,Allison Holker,Singer/Rapper,California,United States,31,21,Eliminated Week 8,7,7,7,7,N/A,7.5,7.5,7,N/A,7,7,8,7,7,8,8,N/A,9,9,9,9,10,10,10,10,9.5,9.5,9,N/A,8.6666,7.6666,7.6666,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Tamar Braxton,Valentin Chmerkovskiy,Singer/Rapper,Maryland,United States,38,21,Withdrew,5,8,7,8,N/A,8,8,8.5,N/A,8,9,8,8,9,9,9,N/A,7,7,8,7,10,10,10,10,9,9,9,N/A,9,9,10,N/A,8.5,8,8,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Nick Carter,Sharna Burgess,Singer/Rapper,New York,United States,35,21,2nd Place,2,8,8,8,N/A,7.5,7.5,7.5,N/A,9,9,9,9,9,9,9,N/A,9,9,8,9,9,10,10,10,9,9,9,N/A,11,11,11,N/A,9,9,9.5,N/A,9,9,9,N/A,10,10,10,N/A
|
||||||
|
Hayes Grier,Emma Slater,Social Media Personality,North Carolina,United States,15,21,Eliminated Week 7,8,7,7,7,N/A,8,7,7.5,N/A,7,8,7,8,9,9,9,N/A,8,8,7,7,8,8,8,8,9,9.5,9.5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Bindi Irwin,Derek Hough,TV Personality,,Australia,17,21,1st Place,1,8,8,8,N/A,8,8,8,N/A,8,8,8,8,9,9,10,N/A,9,9,10,9,10,10,10,10,9.5,10,9.5,N/A,10,9,9,N/A,10,10,10,N/A,10.5,10.5,10.5,N/A,10,10,10,N/A
|
||||||
|
Kim Zolciak-Biermann,Tony Dovolani,TV Personality,Florida,United States,37,21,Withdrew,11,4,4,4,N/A,6.5,6,6,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Paula Deen,Louis van Amstel,TV Personality,Georgia,United States,68,21,Eliminated Week 6,9,5,5,5,N/A,6.5,6,6,N/A,5,5,5,5,6,6,6,N/A,7,7,6,6,6,6,6,6,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Mischa Barton,Artem Chigvintsev,Actor/Actress,,England,30,22,Eliminated Week 3,11,5,5,6,N/A,5,5,5,N/A,6,6,6,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kim Fields,Sasha Farber,Actor/Actress,New York,United States,46,22,Eliminated Week 7,8,7,6,7,N/A,7,6,6,N/A,8,7,7,N/A,8,8,8,8,8,6,6,8,8,8,8,N/A,8.5,8.5,9,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jodie Sweetin,Keo Motsepe,Actor/Actress,California,United States,34,22,Eliminated Week 8,6,7,6,7,N/A,7,7,7,N/A,8,7,8,N/A,7,6,7,7,9,8,9,9,9,8,8,N/A,8,8.5,9,N/A,10,10,10,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Doug Flutie,Karina Smirnoff,Athlete,Maryland,United States,53,22,Eliminated Week 6,9,5,5,5,N/A,7,6,7,N/A,7,6,7,N/A,6,6,6,6,7,7,7,7,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Von Miller,Witney Carson,Athlete,Texas,United States,27,22,Eliminated Week 7,7,8,6,7,N/A,7,6,7,N/A,7,6,7,N/A,8,8,8,8,8,7,7,7,8,8,8,N/A,8.5,8.5,9,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Antonio Brown,Sharna Burgess,Athlete,Florida,United States,27,22,Eliminated Week 9,5,8,6,7,N/A,6,6,7,N/A,7,6,7,N/A,9,8,9,9,7,6,6,7,9,9,9,N/A,8.5,8.5,9,N/A,9,9,9,N/A,9,9,9.5,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Paige VanZant,Mark Ballas,Athlete,Oregon,United States,22,22,2nd Place,2,7,7,7,N/A,8,8,8,N/A,8,7,8,N/A,9,9,9,9,8,8,7,8,9,10,9,N/A,9,9,9.5,N/A,10,9,9,N/A,10,9.5,10,N/A,10,9.6666,10,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Geraldo Rivera,Edyta Sliwinska,Journalist,New York,United States,72,22,Eliminated Week 2,12,5,4,4,N/A,5,4,4,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Nyle DiMarco,Peta Murgatroyd,Model,New York,United States,26,22,1st Place,1,8,7,8,N/A,7,6,7,N/A,8,8,9,N/A,8,8,9,9,9,10,9,9,8,8,9,N/A,9.5,9,9.5,N/A,10,9,10,N/A,9.5,9.5,9.5,N/A,9.6666,9.6666,9.6666,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Ginger Zee,Valentin Chmerkovskiy,News Anchor,California,United States,35,22,3rd Place,3,8,7,8,N/A,7,7,7,N/A,7,7,7,N/A,9,9,9,9,8,8,8,8,8,8,8,N/A,9,9,9.5,N/A,10,10,10,N/A,9,9.5,9.5,N/A,9.3333,9.6666,9.3333,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Wanya Morris,Lindsay Arnold,Singer/Rapper,Pennsylvania,United States,42,22,Eliminated Week 9,4,8,7,8,N/A,8,8,8,N/A,8,8,8,N/A,8,9,9,9,8,7,7,8,10,9,10,N/A,9,9,9.5,N/A,8,8,9,N/A,10,10,10,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Marla Maples,Tony Dovolani,TV Personality,Georgia,United States,52,22,Eliminated Week 4,10,7,7,7,N/A,7,6,7,N/A,7,7,7,N/A,7,7,7,7,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jake T. Austin,Jenna Johnson,Actor/Actress,New York,United States,21,23,Eliminated Week 2,13,5,6,5,6,6,6,6,6,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0
|
||||||
|
Maureen McCormick,Artem Chigvintsev,Actor/Actress,California,United States,60,23,Eliminated Week 7,8,6,5,5,6,7,6,6,7,7,7,7,7,8,8,8,N/A,8,8,8,N/A,8,7,8,8,8.5,8,8,8.5,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0
|
||||||
|
Marilu Henner,Derek Hough,Actor/Actress,Illinois,United States,64,23,Eliminated Week 9,6,7,7,6,7,7,6,7,7,7,7,7,7,7,7,7,N/A,9,9,9,N/A,8,9,9,8,7.5,8,8.5,8,7,8,8,N/A,9,9,9,9,0,0,0,N/A,0,0,0,0
|
||||||
|
Ryan Lochte,Cheryl Burke,Athlete,New York,United States,32,23,Eliminated Week 8,7,6,6,6,6,6,6,6,6,6,7,6,6,7,7,8,N/A,8,8,8,N/A,7,8,8,7,8.5,8,8,8.5,8,9,9,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0
|
||||||
|
Calvin Johnson Jr.,Lindsay Arnold,Athlete,Georgia,United States,30,23,3rd Place,3,7,6,6,7,7,7,7,7,8,8,8,8,8,7,8,N/A,8,8,8,N/A,9,10,9,9,9.5,9,9,9.5,10,10,10,N/A,9,9,10,9,9,9.5,9.5,N/A,9.3333,9.6666,9.6666,9.6666
|
||||||
|
Laurie Hernandez,Valentin Chmerkovskiy,Athlete,New Jersey,United States,16,23,1st Place,1,8,8,7,8,8,8,8,8,7,8,8,8,10,10,10,N/A,8,9,8,N/A,9,9,9,10,8,8.5,9,9,11,11,11,N/A,10,10,10,10,10,10,10,N/A,9.6666,10,9.6666,10
|
||||||
|
Amber Rose,Maksim Chmerkoskiy,Model,Pennsylvania,United States,32,23,Eliminated Week 6,9,6,6,6,6,6,6,6,6,7,6,6,6,8,8,8,N/A,8,8,8,N/A,7,7,7,7,0,0,0,0,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0
|
||||||
|
Rick Perry,Emma Slater,Politician,Texas,United States,66,23,Eliminated Week 3,12,5,5,5,5,6,5,6,5,6,5,6,6,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0
|
||||||
|
James Hinchcliffe,Sharna Burgess,Racing Driver,,Canada,29,23,2nd Place,2,8,8,7,8,7,7,7,8,7,7,8,7,9,9,10,N/A,10,9,10,N/A,10,9,9,10,9.5,9,9,9.5,11.6666,11.6666,11.6666,N/A,9,9,9,9,9.5,10,10,N/A,9.6666,9.6666,9.6666,10
|
||||||
|
Babyface,Allison Holker,Singer/Rapper,Indiana,United States,57,23,Eliminated Week 4,11,7,6,6,7,8,7,7,8,7,6,6,6,6,6,6,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0
|
||||||
|
Vanilla Ice,Witney Carson,Singer/Rapper,Texas,United States,48,23,Eliminated Week 4,10,7,5,6,7,6,7,6,7,6,6,5,6,8,7,8,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,0
|
||||||
|
Jana Kramer,Gleb Savchenko,Singer/Rapper,Michigan,United States,32,23,Eliminated Week 11,4,7,6,6,8,7,8,7,7,6,7,6,7,8,7,8,N/A,9,8,9,N/A,10,10,10,10,8,8.5,9,9,10,10,10,N/A,10,10,10,10,9.5,9.5,10,N/A,8.5,9,9,9
|
||||||
|
Terra Jole,Sasha Farber,TV Personality,Texas,United States,36,23,Eliminated Week 10,5,7,6,6,6,8,7,8,8,8,7,8,7,9,8,8,N/A,9,9,9,N/A,7,8,8,7,8.5,8.5,9,8.5,8,8,8,N/A,10,10,9,9,10,10,10,N/A,0,0,0,0
|
||||||
|
Charo,Keo Motsepe,Actor/Actress,,Spain,66,24,Eliminated Week 3,11,6,5,5,5,6,6,7,6,6,6,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Mr. T,Kym Herjavec,Actor/Actress,Illinois,United States,64,24,Eliminated Week 4,10,5,5,5,5,6,5,6,5,6,6,6,6,7,7,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Heather Morris,Maksim Chmerkoskiy,Actor/Actress,California,United States,30,24,Eliminated Week 6,8,7,7,7,7,8,6,8,8,9,8,8,8,8,9,9,9,8,8,9,9,9,9.5,9,9.5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Nancy Kerrigan,Artem Chigvintsev,Athlete,Massachusetts,United States,47,24,Eliminated Week 7,7,7,7,7,7,7,7,7,7,8,9,8,8,8,9,8,8,9,9,9,9,8.5,8.5,8,8.5,9,9,9,9,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Bonner Bolton,Sharna Burgess,Athlete,Texas,United States,29,24,Eliminated Week 8,5,6,5,5,6,8,5,8,7,6,6,6,6,8,8,8,8,7,8,8,7,8,8.5,7.5,7.5,7.5,7.5,8.5,7.5,7.5,7,7.5,7,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Simone Biles,Sasha Farber,Athlete,Ohio,United States,20,24,Eliminated Week 9,4,8,8,8,8,7,7,7,8,7,8,9,8,9,9,9,9,9,9,10,10,8.5,8.5,8.5,9,10.5,9.5,9.5,9.5,9,9,9,9,10,10,10,10,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
David Ross,Lindsay Arnold,Athlete,Georgia,United States,40,24,2nd Place,2,7,7,7,7,7,6,7,7,8,7,8,8,7,8,8,8,7,7,8,7,7.5,8.5,7.5,7.5,8,8,8,8,8,8,8.5,8,9,8.5,9,8.5,9,9,9.3333,9,N/A,N/A,N/A,N/A
|
||||||
|
Rashad Jennings,Emma Slater,Athlete,Virginia,United States,32,24,1st Place,1,8,7,8,8,8,8,8,8,7,7,7,7,10,9,10,10,7,8,9,8,8.5,9.5,8.5,8.5,9.5,9.5,10.5,9.5,9.5,9,9.5,9.5,9.5,9,10,10,10,9.6666,10,10,N/A,N/A,N/A,N/A
|
||||||
|
Chris Kattan,Witney Carson,Comedian,California,United States,46,24,Eliminated Week 2,12,5,4,4,4,6,5,6,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Erika Jayne,Gleb Savchenko,Singer/Rapper,Georgia,United States,45,24,Eliminated Week 5,9,6,6,6,6,7,7,7,7,6,7,7,6,8,7,7,8,8,8,8,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Normani,Valentin Chmerkovskiy,Singer/Rapper,Georgia,United States,20,24,3rd Place,3,7,6,7,7,8,8,8,8,8,8,9,9,8,7,8,9,10,9,10,10,9,9.5,8,9.5,10.5,10.5,10.5,10.5,10,9.5,10,10,9.5,9.5,9.5,9.5,10,9.6666,9.6666,10,N/A,N/A,N/A,N/A
|
||||||
|
Nick Viall,Peta Murgatroyd,TV Personality,Wisconsin,United States,36,24,Eliminated Week 7,6,6,6,6,6,7,5,7,6,7,6,7,6,8,7,8,7,9,8,9,8,7.5,8,7.5,7.5,8,8,9,9,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Sasha Pieterse,Gleb Savchenko,Actor/Actress,,South Africa,21,25,Eliminated Week 5,10,6,6,6,N/A,8,7.5,7,N/A,7,6,6,N/A,8,8,8,N/A,8,8,8,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Frankie Muniz,Witney Carson,Actor/Actress,New Jersey,United States,31,25,3rd Place,3,7,6,6,N/A,8,7.5,8.5,N/A,7,7,7,N/A,8,8,8,N/A,10,9,10,N/A,7,8,8,8,10,10,10,N/A,9,8.5,9,N/A,8.5,8,9,N/A,9.7777,9.3333,9.7777,10,N/A,N/A,N/A,N/A
|
||||||
|
Jordan Fisher,Lindsay Arnold,Actor/Actress,Alabama,United States,23,25,1st Place,1,8,7,7,N/A,8,8,8,N/A,9,7,9,N/A,10,9,10,N/A,10,10,10,N/A,10,9,10,10,9,9,9,N/A,10,10,10,N/A,9.5,9.5,10,N/A,10,10,10,10,N/A,N/A,N/A,N/A
|
||||||
|
Derek Fisher,Sharna Burgess,Athlete,Arkansas,United States,43,25,Eliminated Week 4,11,6,6,6,N/A,6.5,6,6.5,N/A,7,7,7,N/A,7,8,8,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Nikki Bella,Artem Chigvintsev,Athlete,California,United States,33,25,Eliminated Week 7,8,7,7,6,N/A,6.5,6.5,6.5,N/A,7,7,7,N/A,8,8,8,N/A,9,9,9,N/A,9,9,9,9,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Terrell Owens,Cheryl Burke,Athlete,Alabama,United States,43,25,Eliminated Week 8,6,5,5,5,N/A,6.5,6,7,N/A,7,7,7,N/A,8,8,8,N/A,9,8,8,N/A,9,9,10,9,8,8,8.5,N/A,8.5,8.5,8.5,N/A,0,0,0,N/A,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Victoria Arlen,Valentin Chmerkovskiy,Athlete,New Hamshire,United States,22,25,Eliminated Week 9,5,7,6,6,N/A,7.5,7,7.5,N/A,8,7,7,N/A,9,9,9,N/A,9,9,9,N/A,8,8,7,8,9.5,9.5,9.5,N/A,8,8,8,N/A,9,9.5,9.5,N/A,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Barbara Corcoran,Keo Motsepe,Entrepreneur,New Jersey,United States,68,25,Eliminated Week 2,13,5,4,5,N/A,6,5,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Lindsey Stirling,Mark Ballas,Musician,California,United States,31,25,2nd Place,2,7,8,7,N/A,7.5,7.5,7.5,N/A,9,9,9,N/A,9,8,9,N/A,9,10,9,N/A,10,10,10,10,8.5,8.5,8.5,N/A,9.5,8.5,9,N/A,9.5,9,10,N/A,10,10,10,10,N/A,N/A,N/A,N/A
|
||||||
|
Debbie Gibson,Alan Bersten,Singer/Rapper,New York,United States,47,25,Eliminated Week 2,12,6,5,6,N/A,7,6.5,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Nick Lachey,Peta Murgatroyd,Singer/Rapper,Kentucky,United States,43,25,Eliminated Week 6,9,6,6,6,N/A,7,6,6,N/A,7,7,7,N/A,8,7,7,N/A,7,8,7,N/A,7,6,7,6,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Drew Scott,Emma Slater,TV Personality,,Canada,39,25,Eliminated Week 10,4,6,5,5,N/A,7,6.5,7,N/A,8,7,8,N/A,8,8,8,N/A,9,8,8,N/A,7,7,9,7,9.5,9.5,9.5,N/A,8,7.5,8,N/A,8,8,8,N/A,9,9.5,9.5,9.5,N/A,N/A,N/A,N/A
|
||||||
|
Vanessa Lachey,Maksim Chmerkoskiy,TV Personality,,Philippines,36,25,Eliminated Week 7,7,7,7,7,N/A,8,8,7.5,N/A,7,8,8,N/A,8,8,8,N/A,8,8,8,N/A,9,9,9,9,9,9,9,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Jamie Anderson,Artem Chigvintsev,Athlete,California,United States,27,26,Eliminated Week 1,10,6,7,6,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Johnny Damon,Emma Slater,Athlete,Kansas,United States,44,26,Eliminated Week 1,9,6,6,6,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kareem Abdul-Jabbar,Lindsay Arnold,Athlete,New York,United States,71,26,Eliminated Week 2,8,6,5,6,N/A,7.5,8,7,7,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Arike Ogunbowale,Gleb Savchenko,Athlete,Wisconsin,United States,21,26,Eliminated Week 2,7,7,6,7,N/A,8.5,9.5,8.5,8.5,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jennie Finch Daigle,Keo Motsepe,Athlete,California,United States,37,26,Eliminated Week 3,6,7,7,7,N/A,7.5,8.5,8,7.5,7,8,7,7,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Chris Mazdzer,Witney Carson,Athlete,Massachusetts,United States,29,26,Eliminated Week 3,5,7,7,7,N/A,8.5,8.5,8,8,8,9,7,9,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Mirai Nagasu,Alan Bersten,Athlete,California,United States,25,26,Eliminated Week 3,4,7,8,8,N/A,9,10,9,9,9,9,8,9,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Tonya Harding,Sasha Farber,Athlete,Oregon,United States,47,26,3rd Place,3,8,8,7,N/A,8,9,8,8,8.5,9.5,8.5,8.5,9,9.5,9.5,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Josh Norman,Sharna Burgess,Athlete,South Carolina,United States,30,26,2nd Place,2,8,8,8,N/A,7.5,8,8,8,9.5,9.5,9.5,9.5,9.5,9.5,9.5,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Adam Rippon,Jenna Johnson,Athlete,Pennsylvania,United States,28,26,1st Place,1,8,8,8,N/A,9,9.5,9,9.5,10.5,10.5,9.5,10.5,9.5,9.5,10,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Juan Pablo Di Pace,Cheryl Burke,Actor/Actress,,Argentina,39,27,Eliminated Week 8,5,7,7,8,N/A,9,8,9,N/A,10,10,10,N/A,8,8,8,N/A,10,9,10,N/A,10,10,10,N/A,9.5,9,9.5,N/A,10,10,10,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Evanna Lynch,Keo Motsepe,Actor/Actress,,Ireland,27,27,3rd Place,3,7,5,6,N/A,8,8,8,N/A,9,9,9,N/A,8,8,8,N/A,8,8,8,N/A,10,9,10,N/A,10,9.5,10,N/A,10,9.5,9.5,N/A,10,10,10,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Nancy McKeon,Valentin Chmerkovskiy,Actor/Actress,New York,United States,52,27,Eliminated Week 3,11,6.5,6.5,6.5,N/A,7,6.5,7,N/A,8,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
John Schneider,Emma Slater,Actor/Actress,New York,United States,58,27,Eliminated Week 7,8,7,5,6,N/A,7.5,6.5,7.5,N/A,7,7,7,N/A,7,7,7,N/A,8,8,8,N/A,6,7,6,N/A,9,9,9,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Milo Manheim,Witney Carson,Actor/Actress,California,United States,17,27,2nd Place,2,7,6,7,N/A,9,8,9,N/A,9,9,9,N/A,10,9,10,N/A,9,8,10,N/A,10,10,10,N/A,10,9,10,N/A,9,9,9.5,N/A,10,10,10,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Danelle Umstead,Artem Chigvintsev,Athlete,New Mexico,United States,46,27,Eliminated Week 2,12,6,6,6,N/A,6.5,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Mary Lou Retton,Sasha Farber,Athlete,West Virginia,United States,50,27,Eliminated Week 6,9,6.5,7,6.5,N/A,8,7.5,7.5,N/A,8,8,8,N/A,9,8,9,N/A,9,8,8,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
DeMarcus Ware,Lindsay Arnold,Athlete,Alabama,United States,36,27,Eliminated Week 7,7,8,7,8,N/A,8,7.5,8,N/A,9,8,9,N/A,7,7,8,N/A,9,8,9,N/A,8,9,9,N/A,9,8.5,9,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Nikki Glaser,Gleb Savchenko,Comedian,Ohio,United States,34,27,Eliminated Week 1,13,6,5.5,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Bobby Bones,Sharna Burgess,Radio Personality,Arkansas,United States,38,27,1st Place,1,7,6,7,N/A,7,6,6.5,N/A,8,7,8,N/A,7,6,7,N/A,7,7,7,N/A,8,7,7,N/A,9,8.5,9,N/A,7.5,7.5,7.5,N/A,9,9,9,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Tinashe,Brandon Armstrong,Singer/Rapper,Kentucky,United States,25,27,Eliminated Week 4,10,8,7,8,N/A,9,8,9,N/A,9,9,9,N/A,9,8,9,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Alexis Ren,Alan Bersten,Social Media Personality,California,United States,21,27,4th Place,4,7,7.5,7.5,N/A,8,8.5,8,N/A,9,8,9,N/A,8,8,9,N/A,10,9,10,N/A,9,9,9,N/A,9,9,9.5,N/A,9.5,9.5,10,N/A,9.5,9.5,9.5,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Joe Amabile,Jenna Johnson,TV Personality,Illinois,United States,32,27,Eliminated Week 8,6,5,4,5,N/A,5.5,6,6,N/A,6,6,6,N/A,5,5,5,N/A,6,5,6,N/A,8,7,7,N/A,8,7.5,8,N/A,8,7.5,7.5,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Kate Flannery,Pasha Pashkov,Actor/Actress,Pennsylvania,United States,55,28,Eliminated Week 8,7,5,5,5,N/A,7,7,7,N/A,8,8,8,N/A,7,7,6,6,8,8,8,N/A,9,9,9,N/A,8,8,8,N/A,8.6666,8.6666,8.6666,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
James Van Der Beek,Emma Slater,Actor/Actress,Connecticut,United States,42,28,Eliminated Week 10,5,7,7,7,N/A,7,6,7,N/A,8,7,8,N/A,7,7,7,7,9,8,9,N/A,9,9,9,N/A,9,9,9,N/A,10.6666,10.6666,10.6666,N/A,9,8.5,9.5,9,8.5,8.5,8.5,N/A,0,0,0,N/A
|
||||||
|
Kel Mitchell,Witney Carson,Actor/Actress,Illinois,United States,41,28,2nd Place,2,6,5,5,N/A,7,6,7,N/A,7,6,7,N/A,8,8,8,8,9,8,9,N/A,9,8,9,N/A,8.5,8.5,8.5,N/A,9.6666,9.6666,10.6666,N/A,9,9,9.5,9.5,9.5,9.5,9.5,N/A,10,9.5,10,N/A
|
||||||
|
Ray Lewis,Cheryl Burke,Athlete,Florida,United States,44,28,Withdrew,11,5,5,5,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Lamar Odom,Peta Murgatroyd,Athlete,New York,United States,39,28,Eliminated Week 4,10,5,3,3,N/A,4,4,4,N/A,4,4,4,N/A,5,7,4,4,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Sailor Brinkley-Cook,Valentin Chmerkovskiy,Model,New York,United States,21,28,Eliminated Week 6,9,6,6,6,N/A,6,6,6,N/A,7,8,8,N/A,7,8,8,8,8,8,8,N/A,9,9,9,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Sean Spicer,Lindsay Arnold,Politician,New York,United States,48,28,Eliminated Week 9,6,4,4,4,N/A,6,5,5,N/A,5,5,5,N/A,5,6,5,5,7,6,6,N/A,7,7,7,N/A,7,7,7,N/A,7,7,6,N/A,6.5,6,6.5,6,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Mary Wilson,Brandon Armstrong,Singer/Rapper,Nevada,United States,75,28,Eliminated Week 2,12,6,5,6,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Lauren Alaina,Gleb Savchenko,Singer/Rapper,Georgia,United States,24,28,4th Place,4,7,6,6,N/A,6,6,7,N/A,6,7,7,N/A,8,8,8,8,8,7,8,N/A,9,8,9,N/A,9,9,9,N/A,8,8,8,N/A,9,9,8.5,8.5,9,9,9,N/A,9.5,9.5,9.5,N/A
|
||||||
|
Ally Brooke,Sasha Farber,Singer/Rapper,Texas,United States,26,28,3rd Place,3,5,5,6,N/A,7,6,7,N/A,8,8,8,N/A,8,8,8,8,9,9,9,N/A,8,9,8,N/A,9,9,9,N/A,10,10,10,N/A,10,10,10,10,10,9.5,10,N/A,10,10,10,N/A
|
||||||
|
Karamo Brown,Jenna Johnson,TV Personality,Texas,United States,38,28,Eliminated Week 7,8,6,5,6,N/A,7,5,7,N/A,5,5,6,N/A,7,7,7,7,7,7,7,N/A,9,8,8,N/A,8.5,8,8,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Hannah Brown,Alan Bersten,TV Personality,Alabama,United States,24,28,1st Place,1,7,7,6,N/A,8,8,8,N/A,7,7,7,N/A,8,8,8,8,9,7,9,N/A,8,8,8,N/A,8.5,9,8.5,N/A,10.6666,9.6666,10.6666,N/A,9,8.5,9,9,9,9,9,N/A,10,9.5,9.5,N/A
|
||||||
|
Anne Heche,Keo Motsepe,Actor/Actress,Ohio,United States,51,29,Eliminated Week 4,13,6,6,6,N/A,6,6,6,N/A,5,5,5,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Jesse Metcalfe,Sharna Burgess,Actor/Actress,California,United States,41,29,Eliminated Week 5,12,6,6,6,N/A,7,7,6,N/A,7,6,7,N/A,7,7,7,N/A,7,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Chrishell Stause,Gleb Savchenko,Actor/Actress,Kentucky,United States,39,29,Eliminated Week 8,8,4,5,4,N/A,6,6,6,N/A,7,7,8,N/A,7,8,7,N/A,6,6,7,N/A,8,8,8,N/A,9,9,8,N/A,9,9,9,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Skai Jackson,Alan Bersten,Actor/Actress,New York,United States,18,29,Eliminated Week 10,5,7,7,7,N/A,5,5,5,N/A,6,6,6,N/A,10,9,9,N/A,8,8,8,N/A,6,6,6,N/A,9,9,9,N/A,8.6666,9.6666,8.6666,N/A,9.6666,9.6666,9.6666,N/A,9.5,9.5,9.5,N/A,0,0,0,N/A
|
||||||
|
Justina Machado,Sasha Farber,Actor/Actress,Illinois,United States,48,29,4th Place,4,7,7,7,N/A,7,7,7,N/A,7,6,6,N/A,8,8,8,N/A,8,8,8,N/A,9,9,9,N/A,9,9,8,N/A,9.6666,9.6666,9.6666,N/A,8.6666,8.6666,8.6666,N/A,9.5,10,9.5,N/A,10,10,10,N/A
|
||||||
|
Charles Oakley,Emma Slater,Athlete,Ohio,United States,56,29,Eliminated Week 2,15,4,4,4,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Veron Davis,Peta Murgatroyd,Athlete,Washington D.C.,United States,36,29,Eliminated Week 6,11,5,6,6,N/A,6,6,6,N/A,8,7,7,N/A,8,7,7,N/A,7,7,7,N/A,7,7,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Johnny Weir,Britt Stewart,Athlete,Pennsylvania,United States,36,29,Eliminated Week 10,6,6,6,6,N/A,6,6,6,N/A,8,8,8,N/A,8,8,8,N/A,10,10,9,N/A,7,8,7,N/A,9,9,9,N/A,10,10,10,N/A,10,10,10,N/A,9.5,9.5,9.5,N/A,0,0,0,N/A
|
||||||
|
Nev Schulman,Jenna Johnson,Producer,New York,United States,35,29,2nd Place,2,7,7,6,N/A,7,7,7,N/A,8,8,8,N/A,8,8,8,N/A,8,9,9,N/A,9,9,8,N/A,10,10,10,N/A,10,10,10,N/A,9.6666,9.6666,9.6666,N/A,10,10,10,N/A,10,10,10,N/A
|
||||||
|
AJ McLean,Cheryl Burke,Singer/Rapper,Florida,United States,42,29,Eliminated Week 9,7,6,6,6,N/A,7,6,6,N/A,7,7,7,N/A,8,8,8,N/A,8,8,8,N/A,9,9,9,N/A,9,8,9,N/A,9,9,9,N/A,7.6666,8.6666,8.6666,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Nelly,Daniella Karagach,Singer/Rapper,Texas,United States,45,29,3rd Place,3,5,5,6,N/A,6,6,6,N/A,6,6,6,N/A,7,7,7,N/A,8,8,8,N/A,8,8,8,N/A,9,9,9,N/A,8,8,8,N/A,8,8,8,N/A,9.5,9,9.5,N/A,9.5,9.5,9.5,N/A
|
||||||
|
Kaitlyn Bristowe,Artem Chigvintsev,TV Personality,,Canada,35,29,1st Place,1,6,7,7,N/A,7,8,7,N/A,8,7,8,N/A,9,8,8,N/A,9,9,9,N/A,9,9,9,N/A,7,9,8,N/A,9,10,9,N/A,10,10,10,N/A,10,10,10,N/A,10,10,10,N/A
|
||||||
|
Carole Baskin,Pasha Pashkov,TV Personality,Texas,United States,59,29,Eliminated Week 3,14,4,4,3,N/A,6,5,5,N/A,5,4,3,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Monica Aldama,Valentin Chmerkovskiy,TV Personality,Alabama,United States,48,29,Eliminated Week 7,10,6,7,6,N/A,5,6,5,N/A,7,7,7,N/A,8,8,8,N/A,9,9,8,N/A,9,9,9,N/A,7,8,7,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Jeannie Mai,Brandon Armstrong,TV Personality,California,United States,41,29,Withdrew,9,6,6,6,N/A,6,6,6,N/A,7,7,8,N/A,7,7,7,N/A,8,8,8,N/A,8,8,9,N/A,8,9,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Martin Kove,Britt Stewart,Actor/Actress,New York,United States,75,30,Eliminated Week 2,15,4,3,3,3,4,3,4,4,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Brian Austin Green,Sharna Burgess,Actor/Actress,California,United States,48,30,Eliminated Week 4,13,6,6,6,6,6,5,6,6,7,6,6,N/A,6.5,6,7,6.5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Melora Hardin,Artem Chigvintsev,Actor/Actress,Texas,United States,54,30,Eliminated Week 9,6,7,6,7,6,7,6,7,7,8,7,8,N/A,9,9.5,9,9,9,9,9,9,8,9,8,9,9.5,9.5,9.5,9.5,10.5,10.5,10.5,10.5,8.5,9.5,9,9,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Matt James,Lindsay Arnold,Athlete,North Carolina,United States,29,30,Eliminated Week 4,12,6,6,6,6,5,5,6,6,7,6,7,N/A,7,6.5,7.5,7.5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
"Mike ""The Miz"" Mizanin",Witney Carson,Athlete,Ohio,United States,40,30,Eliminated Week 7,9,6,6,6,6,7,5,7,7,7,7,8,N/A,8.5,7.5,8.5,8,8,8,8,8,9,8,9,8,8,8,8,8,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Suni Lee,Sasha Farber,Athlete,Minnesota,United States,18,30,Eliminated Week 9,5,7,7,7,7,7,7,7,7,7,7,7,N/A,8.5,8,9,8.5,9,9,9,9,9,9,9,9,8.25,8.25,8.25,9.25,10.5,10.5,10.5,10.5,10,9,9.5,9,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
Iman Shumpert,Daniella Karagach,Athlete,Illinois,United States,31,30,1st Place,1,7,4,5,5,7,6,6,6,7,6,6,N/A,8.5,6.5,8.5,8,7,7,7,7,10,10,10,10,9.5,7.5,8.5,8.5,9,8,9,9,9.5,9,10,9,10,10,10,10,N/A,N/A,N/A,N/A
|
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|
Cody Rigsby,Cheryl Burke,Fitness Instructor,California,United States,34,30,3rd Place,3,6,6,6,6,6,6,6,6,6,6,6,N/A,7.5,6.5,7.5,7.5,8,8,8,8,9,9,9,9,8.5,8.5,9.5,9.5,10,10,9,9,9,9,8.5,9,9.5,9.5,9.5,9.5,N/A,N/A,N/A,N/A
|
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|
Melanie C,Gleb Savchenko,Singer/Rapper,,England,47,30,Eliminated Week 5,11,7,7,6,7,7,7,8,8,7,7,8,N/A,8,8,8.5,8.5,9,9,9,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
Jimmie Allen,Emma Slater,Singer/Rapper,Delaware,United States,36,30,Eliminated Week 8,7,6,5,6,5,7,6,7,7,6,6,8,N/A,8.5,7.5,8.5,8,8,8,9,9,10,9,10,9,9,9,10,10,8,8,8,8,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
Olivia Jade,Valentin Chmerkovskiy,Social Media Personality,California,United States,21,30,Eliminated Week 8,8,7,6,6,6,7,6,7,7,8,8,8,N/A,8,9,9,9,9,9,9,9,9,9,9,9,11,9,11,11,9,9,9,9,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
JoJo Siwa,Jenna Johnson,Social Media Personality,Nebraska,United States,18,30,2nd Place,2,8,7,7,7,8,8,7,8,8,8,8,N/A,8.5,8.5,9,9,10,10,10,10,10,10,10,10,10,9,10,10,10.5,9.5,10.5,10.5,10,10,10,10,10,10,10,10,N/A,N/A,N/A,N/A
|
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|
Christine Chiu,Pasha Pashkov,TV Personality,,Taiwan China,38,30,Eliminated Week 3,14,6,7,6,6,6,6,6,6,7,7,7,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Kenya Moore,Brandon Armstrong,TV Personality,Michigan,United States,50,30,Eliminated Week 6,10,7,6,6,7,6,6,6,6,7,7,7,N/A,7,7,8,7.5,9,9,9,9,8,8,8,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Amanda Kloots,Alan Bersten,TV Personality,Ohio,United States,39,30,4th Place,4,7,7,7,7,8,8,8,8,8,8,8,N/A,8.5,8.5,8.5,8.5,10,10,9,10,9,9,10,10,8.25,8.25,8.25,9.25,10.5,10.5,10.5,10.5,9.5,10,10,10,9.5,10,9.5,10,N/A,N/A,N/A,N/A
|
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|
Jason Lewis,Peta Murgatroyd,Actor/Actress,California,United States,51,31,Eliminated Week 1,16,5,4,4,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
Cheryl Ladd,Louis van Amstel,Actor/Actress,South Dakota,United States,71,31,Eliminated Week 3,14,6,5,5,5,5,6,5,5,6,6,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Selma Blair,Sasha Farber,Actor/Actress,Michigan,United States,50,31,Withdrew,12,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,10,10,10,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Joseph Baena,Daniella Karagach,Actor/Actress,California,United States,24,31,Eliminated Week 5,11,6,5,6,6,6,6,6,6,8,7,7,7,7,7,7,7,9,8.5,9,8.5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Trevor Donovan,Emma Slater,Actor/Actress,California,United States,39,31,Eliminated Week 9,6,5,5,5,6,7,7,8,8,7,6,7,7,7,7,7,7,9,9,9,9,9,8,8,9,10,9,10,10,9.25,10.25,9.25,10.25,8.5,8,8,8,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Daniel Durant,Britt Stewart,Actor/Actress,Michigan,United States,32,31,Eliminated Week 9,5,7,6,7,7,7,7,7,8,8,7,8,8,7,7,7,8,8.125,8.125,8.625,8.625,9,8,8,9,9,8.5,9.5,9.5,10,9,10,10,9,8,9,8.5,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Wayne Brady,Witney Carson,Comedian,Georgia,United States,50,31,3rd Place,3,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,10.125,10.125,10.125,10.625,8,9,8,9,9.5,9,9.5,10,11.25,11.25,11.25,11.25,9.5,9,9,9,9.5,9.5,9.5,9.5,N/A,N/A,N/A,N/A
|
||||||
|
Sam Champion,Cheryl Burke,News Anchor,Kentucky,United States,61,31,Eliminated Week 4,13,5,5,5,5,6,6,7,7,6,6,6,7,7,6,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
||||||
|
Jessie James Decker,Alan Bersten,Singer/Rapper,,Italy,34,31,Eliminated Week 6,10,5,5,5,5,7,6,6,6,6,6,7,7,8,7,8,8,8.875,8.875,9.375,8.875,8,8,8,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
Jordin Sparks,Brandon Armstrong,Singer/Rapper,Arizona,United States,32,31,Eliminated Week 7,9,7,6,6,7,7,6,7,7,7,7,7,8,9,8,8,9,8.875,8.375,8.875,9.375,9,8,8,9,8.5,9,8.5,8.5,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
Heidi D'Amelio,Artem Chigvintsev,Social Media Personality,Louisiana,United States,50,31,Eliminated Week 8,8,6,6,6,6,7,7,7,7,8,8,8,8,9,8,8,9,9.625,10.125,9.625,9.625,9,9,9,9,8.5,9,9,8.5,9,8,9,9,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
Charli D'Amelio,Mark Ballas,Social Media Personality,Connecticut,United States,18,31,1st Place,1,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10.375,10.375,10.375,10.875,10,10,10,10,9.5,9.5,10,10,11.25,11.25,11.25,11.25,10,10,10,10,10,10,10,10,N/A,N/A,N/A,N/A
|
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|
Teresa Giudice,Pasha Pashkov,TV Personality,New Jersey,United States,50,31,Eliminated Week 2,15,5,5,5,5,6,5,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
Vinny Guadagnino,Koko Iwasaki,TV Personality,New York,United States,34,31,Eliminated Week 8,7,4,4,5,4,7,6,7,7,6,5,6,6,8,7,7,7,7.75,7.75,7.75,7.75,7,7,7,7,7.5,8,8,8,8,7,7,7,0,0,0,0,0,0,0,0,N/A,N/A,N/A,N/A
|
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|
Shangela,Gleb Savchenko,TV Personality,Texas,United States,40,31,4th Place,4,7,7,7,7,7,7,7,7,8,7,7,8,8,8,8,8,9.75,9.25,9.75,9.75,9,9,9,9,9,9.5,9,9,11.25,10.25,10.25,10.25,9.5,9,9,9,9.5,9.5,9.5,9.5,N/A,N/A,N/A,N/A
|
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|
Gabby Windey,Valentin Chmerkovskiy,TV Personality,Illinois,United States,31,31,2nd Place,2,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,10.25,10.25,10.25,10.25,9,9,9,10,9,9,8.5,9,10,10,10,10,10,10,10,10,10,10,10,10,N/A,N/A,N/A,N/A
|
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|
Jamie Lynn Spears,Alan Bersten,Actor/Actress,Mississippi,United States,32,32,Eliminated Week 2,13,5,5,5,N/A,6,5,5,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
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|
Mira Sorvino,Gleb Savchenko,Actor/Actress,New York,United States,55,32,Eliminated Week 5,10,6,5,6,N/A,6,6,6,N/A,6,6,7,7,7,7,7,N/A,7,8,7,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
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|
Barry Williams,Peta Murgatroyd,Actor/Actress,California,United States,68,32,Eliminated Week 8,7,6,5,5,N/A,5,5,5,N/A,6,6,7,6,6,6,6,N/A,8,8,8,N/A,8.25,7.25,8.25,8.25,8,8,8.5,8,8.75,8.75,8.75,8.75,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
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|
Alyson Hannigan,Sasha Farber,Actor/Actress,Washington D.C.,United States,49,32,5th Place,5,5,4,4,N/A,7,6,6,N/A,6,6,6,6,6,6,6,N/A,7,7,7,N/A,7.5,7.5,7.5,8.5,8.5,8.5,9.5,8.5,8.75,8.75,9.75,8.75,7.75,7.75,8.75,7.75,8.5,8.5,8.5,N/A,9,8.5,9,N/A
|
||||||
|
Xochitl Gomez,Valentin Chmerkovskiy,Actor/Actress,California,United States,17,32,1st Place,1,6,6,6,N/A,8,8,8,N/A,8,8,8,8,9,9,9,N/A,9,10,9,N/A,10.25,10.25,11.25,10.25,9.5,9,9.5,9,10,10,10,10,9.75,9.75,10.75,10.75,10,10,10,N/A,10,10,10,N/A
|
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|
Adrian Peterson,Britt Stewart,Athlete,Texas,United States,38,32,Eliminated Week 4,11,6,6,6,N/A,5,5,5,N/A,5,5,6,6,7,7,7,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
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|
Matt Walsh,Koko Iwasaki,Comedian,Illinois,United States,58,32,Eliminated Week 1,14,4,4,4,N/A,0,0,0,N/A,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
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|
Tyson Beckford,Jenna Johnson,Model,New York,United States,52,32,Eliminated Week 3,12,4,4,4,N/A,6,6,6,N/A,5,5,5,5,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Jason Mraz,Daniella Karagach,Singer/Rapper,Virginia,United States,46,32,2nd Place,2,7,7,7,N/A,8,8,8,N/A,9,8,9,8,8,8,8,N/A,9,9,9,N/A,9.75,9.75,9.75,9.75,8.5,8.5,10,9,8,9,8,8,10.75,10.75,10.75,10.75,9.5,9.5,9.5,N/A,10,10,10,N/A
|
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|
Lele Pons,Brandon Armstrong,Social Media Personality,,Venezuela,27,32,Eliminated Week 7,8,6,7,6,N/A,7,7,7,N/A,7,6,7,7,7,7,8,N/A,8,8,8,N/A,9,9,9,10,9.5,9,9,9,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
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|
Harry Jowsey,Rylee Arnold,TV Personality,,Australia,26,32,Eliminated Week 9,6,4,4,4,N/A,5,5,5,N/A,6,5,7,6,7,7,7,N/A,6,6,6,N/A,7.25,7.25,7.25,7.25,8,8,8,8,7,7,7,7,8,7,8,7,0,0,0,N/A,0,0,0,N/A
|
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|
Mauricio Umansky,Emma Slater,TV Personality,,Mexico,53,32,Eliminated Week 6,9,5,5,5,N/A,4,4,4,N/A,7,5,6,5,7,6,6,N/A,8,8,8,N/A,8.5,8.5,7.5,8.5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
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|
Charity Lawson,Artem Chigvintsev,TV Personality,Georgia,United States,27,32,4th Place,4,7,7,8,N/A,7,7,7,N/A,8,8,8,8,8,8,8,N/A,10,9,9,N/A,9.75,9.75,8.75,9.75,9,9,9,9,9.75,9.75,9.75,9.75,9,10,9,10,10,9.5,10,N/A,9.5,10,10,N/A
|
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|
Ariana Madix,Pasha Pashkov,TV Personality,Florida,United States,38,32,3rd Place,3,7,7,7,N/A,6,7,7,N/A,9,8,9,8,8,9,8,N/A,8,8,8,N/A,10,10,10,11,9,9,10,9,9.75,10.75,10.75,10.75,9,9,9,10,9.5,9.5,10,N/A,9.5,10,10,N/A
|
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|
Tori Spelling,Pasha Pashkov,Actor/Actress,California,United States,51,33,Eliminated Week 2,12,6,6,5,N/A,7,6,6,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
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|
Eric Roberts,Britt Stewart,Actor/Actress,California,United States,68,33,Eliminated Week 3,10,5,5,5,N/A,6,4,5,N/A,6.5,5.5,5.5,5.5,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
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|
Reginald VelJohnson,Emma Slater,Actor/Actress,New York,United States,74,33,Eliminated Week 3,10,6,5,5,N/A,6,4,5,N/A,6,5,5,5,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Chandler Kinney,Brandon Armstrong,Actor/Actress,California,United States,24,33,3rd Place,3,8,7,8,N/A,8,8,8,N/A,8.5,8.5,9,8.5,8,8,8,8,9,9,9,N/A,10,11,11,N/A,10,10,10,N/A,9.5,9.5,10,N/A,10,10,10,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
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|
Dwight Howard ,Daniella Karagach,Athlete,Georgia,United States,38,33,Eliminated Week 7,6,8,7,7,N/A,8,7,7,N/A,7,6.5,6,6.5,7,8,7,7,8,8,8,N/A,11,10,10,N/A,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Danny Amendola,Witney Carson,Athlete,Texas,United States,39,33,5th Place,5,7,6,7,N/A,7,7,7,N/A,7.5,7.5,8,7,9,9,9,9,8.5,8.5,8.5,N/A,10,9,9,N/A,8.5,8.5,8.5,N/A,9,9,9,N/A,9,9.5,9.5,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Stephen Nedoroscik,Rylee Arnold,Athlete,Florida,United States,26,33,4th Place,4,7,7,7,N/A,8,7,7,N/A,8,8,7.5,7.5,8,9,8,8,8,8,8,N/A,10,9,9,N/A,9.5,9,8.5,N/A,9,8.5,9,N/A,10,9.5,10,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Ilona Maher,Alan Bersten,Athlete,California,United States,28,33,2nd Place,2,6,6,6,N/A,7,7,7,N/A,7.5,7,7,6.5,8,8,8,8,9,8.5,8.5,N/A,8,8,8,N/A,9,8.5,8.5,N/A,9.5,9.5,9.5,N/A,9.5,9.5,9.5,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
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|
Anna Delvey,Ezra Sosa,Con artist,,Russia,33,33,Eliminated Week 2,12,6,6,6,N/A,6,5,6,N/A,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Brooks Nader,Gleb Savchenko,Model,New York,United States,27,33,Eliminated Week 4,9,6,6,6,N/A,7,6,7,N/A,7.5,7.5,8,7.5,8,8,8,8,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Phaedra Parks,Val Chmerkovskiy,TV Personality,Georgia,United States,51,33,Eliminated Week 5,8,7,6,6,N/A,7,7,7,N/A,7.5,7.5,6,7,8,8,8,8,8,8,8,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Jenn Tran,Sasha Farber,TV Personality,Florida,United States,26,33,Eliminated Week 6,7,7,6,6,N/A,6,6,7,N/A,8,8,7,8,8,8,8,8,8.5,8.5,8.5,N/A,11,10,10,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Joey Graziadei,Jenna Johnson,TV Personality,Louisiana,United States,29,33,1st Place,1,7,7,7,N/A,8,7,7,N/A,9,9,8.5,8.5,9,9,9,9,8.5,8.5,9,N/A,11,10,11,N/A,9,9.5,9,N/A,9.5,10,9.5,N/A,9.5,10,10,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A,N/A
|
||||||
|
Corey Feldman,Jenna Johnson,Actor/Actress,California,United States,54,34,Eliminated Week 2,13,4,5,N/A,N/A,5,5,5,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Danielle Fishel,Pasha Pashkov,Actor/Actress,California,United States,44,34,Eliminated Week 8,8,6,6,N/A,N/A,7,6,6,N/A,7,7,7,N/A,7,7,7,N/A,7,7,7,8,9,9,9,9,8.5,8.5,8.5,9.5,9.5,9,9.5,9,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Elaine Hendrix,Alan Bersten (Rashad Jennings week 9),Actor/Actress,Georgia,United States,55,34,5th Place,5,6,6,N/A,N/A,7,7,7,N/A,7,7,7,N/A,8,8,8,N/A,8,7,7,8,9,9,9,9,8,8,8,8,9,9.5,10,9,9,9,9,9,9.5,9.5,9.5,N/A,9.6666,9.6666,9.6666,N/A
|
||||||
|
Baron Davis,Britt Stewart,Athlete,California,United States,46,34,Eliminated Week 2,13,5,5,N/A,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Jordan Chiles,Ezra Sosa (Apolo Anton Ohno week 9),Athlete,Texas,United States,24,34,3rd Place,3,5,5,N/A,N/A,8,7,7,N/A,8,8,8,N/A,8,8,8,N/A,8,8,8,8,10,9,10,10,10,9,9,10,10,9.5,10,9.5,9.5,10.5,9.5,9.5,9.5,9.5,9.5,N/A,9.6666,10,10,N/A
|
||||||
|
Andy Richter,Emma Slater/Kaitlyn Bristowe (week 9),Comedian,California,United States,59,34,Eliminated Week 9,7,5,4,N/A,N/A,6,5,5,N/A,6,6,6,N/A,6,6,6,N/A,6,6,6,6,7,6,7,7,7.25,7.25,7.25,7.25,8,8.5,9.5,8,8,7,7,7,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Robert Irwin,Witney Carson (Xoshitl Gomez week 9),Conservationist,,Australia,22,34,1st Place,1,8,7,N/A,N/A,8,7,7,N/A,8,7,7,N/A,7,7,8,N/A,8,9,9,9,9,9,9,9,10.75,9.75,9.75,10.75,9,9.5,10,9.5,10.5,10.5,10.5,10.5,10,9.5,10,N/A,9.6666,10,10,N/A
|
||||||
|
Hilaria Baldwin,Gleb Savchenko,Entrepreneur,New York,United States,41,34,Eliminated Week 4,11,7,7,N/A,N/A,7,7,7,N/A,7,8,7,N/A,8,7,8,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Lauren Jauregui,Brandon Armstrong,Singer/Rapper,Florida,United States,29,34,Eliminated Week 3,12,7,6,N/A,N/A,7,7,7,N/A,6,6,6,N/A,0,0,0,N/A,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Scott Hoying,Rylee Arnold,Singer/Rapper,California,United States,34,34,Eliminated Week 6,10,5,5,N/A,N/A,6,6,6,N/A,8,7,7,N/A,7,7,7,N/A,7,8,7,8,7,7,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Alix Earle,Val Chmerkovskiy (Joey Graziadei week 9),Social media personality,Florida,United States,24,34,2nd Place,2,7,6,N/A,N/A,7,7,7,N/A,8,7,8,N/A,8,8,8,N/A,8,9,9,9,9,9,8,9,11,11,10,11,9.5,10,9.5,9.5,10,10,10,10,9.5,10,9.5,N/A,10,10,10,N/A
|
||||||
|
Jen Affleck,Jan Ravnik,TV Personality,Utah,United States,26,34,Eliminated Week 7,9,6,6,N/A,N/A,8,7,7,N/A,7,6,6,N/A,7,8,8,N/A,8,7,7,7,8,8,8,8,8.5,8.5,8.5,8.5,0,0,0,0,0,0,0,0,0,0,0,N/A,0,0,0,N/A
|
||||||
|
Whitney Leavitt,Mark Ballas,TV Personality,Utah,United States,32,34,Eliminated Week 10,6,7,8,N/A,N/A,7,7,8,N/A,8,8,8,N/A,9,8,8,N/A,9,8,8,8,10,9,10,10,10.25,10.25,10.25,11.25,9.5,10,10,10,10.5,10.5,10.5,10.5,9,10,10,N/A,0,0,0,N/A
|
||||||
|
Dylan Efron,Daniella Karagach (Rumer Willis week 9),TV Personality,California,United States,33,34,4th Place,4,5,5,N/A,N/A,7,6,7,N/A,7,8,8,N/A,7,8,8,N/A,9,9,9,9,8,8,8,8,9.75,9.75,8.75,9.75,9,9.5,10,9.5,10.5,10.5,10.5,10.5,9,9,9.5,N/A,9.6666,9.6666,10,N/A
|
||||||
|
BIN
2026_MCM-ICM_Problems/Contest_AI_Policy.pdf
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BIN
2026_MCM-ICM_Problems/Contest_AI_Policy.pdf
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2026_MCM-ICM_Problems/MCM-ICM_SubProcess.pdf
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2026_MCM-ICM_Problems/MCM-ICM_SubProcess.pdf
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2026_MCM-ICM_Problems/MCM-ICM_Summary.docx
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76
2026_MCM-ICM_Problems/MCM-ICM_Summary.tex
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76
2026_MCM-ICM_Problems/MCM-ICM_Summary.tex
Normal file
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|
|||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
%% MCM/ICM LaTeX Template %%
|
||||||
|
%% 2026 MCM/ICM %%
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\documentclass[12pt]{article}
|
||||||
|
\usepackage{geometry}
|
||||||
|
\geometry{left=1in,right=0.75in,top=1in,bottom=1in}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
% Replace ABCDEF in the next line with your chosen problem
|
||||||
|
% and replace 1111111 with your Team Control Number
|
||||||
|
\newcommand{\Problem}{ABCDEF}
|
||||||
|
\newcommand{\Team}{1111111}
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
|
||||||
|
\usepackage{newtxtext}
|
||||||
|
\usepackage{amsmath,amssymb,amsthm}
|
||||||
|
\usepackage{newtxmath} % must come after amsXXX
|
||||||
|
|
||||||
|
\usepackage[pdftex]{graphicx}
|
||||||
|
\usepackage{xcolor}
|
||||||
|
\usepackage{fancyhdr}
|
||||||
|
\lhead{Team \Team}
|
||||||
|
\rhead{}
|
||||||
|
\cfoot{}
|
||||||
|
|
||||||
|
\newtheorem{theorem}{Theorem}
|
||||||
|
\newtheorem{corollary}[theorem]{Corollary}
|
||||||
|
\newtheorem{lemma}[theorem]{Lemma}
|
||||||
|
\newtheorem{definition}{Definition}
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\begin{document}
|
||||||
|
\graphicspath{{.}} % Place your graphic files in the same directory as your main document
|
||||||
|
\DeclareGraphicsExtensions{.pdf, .jpg, .tif, .png}
|
||||||
|
\thispagestyle{empty}
|
||||||
|
\vspace*{-16ex}
|
||||||
|
\centerline{\begin{tabular}{*3{c}}
|
||||||
|
\parbox[t]{0.3\linewidth}{\begin{center}\textbf{Problem Chosen}\\ \Large \textcolor{red}{\Problem}\end{center}}
|
||||||
|
& \parbox[t]{0.3\linewidth}{\begin{center}\textbf{2026\\ MCM/ICM\\ Summary Sheet}\end{center}}
|
||||||
|
& \parbox[t]{0.3\linewidth}{\begin{center}\textbf{Team Control Number}\\ \Large \textcolor{red}{\Team}\end{center}} \\
|
||||||
|
\hline
|
||||||
|
\end{tabular}}
|
||||||
|
%%%%%%%%%%% Begin Summary %%%%%%%%%%%
|
||||||
|
% Enter your summary here replacing the (red) text
|
||||||
|
% Replace the text from here ...
|
||||||
|
\begin{center}
|
||||||
|
\textcolor{red}{%
|
||||||
|
Use this template to begin typing the first page (summary page) of your electronic report. This \newline
|
||||||
|
template uses a 12-point Times New Roman font. Submit your paper as an Adobe PDF \newline
|
||||||
|
electronic file (e.g. 1111111.pdf), typed in English, with a readable font of at least 12-point type. \\[2ex]
|
||||||
|
Do not include the name of your school, advisor, or team members on this or any page. \\[2ex]
|
||||||
|
Be sure to change the control number and problem choice above. \\
|
||||||
|
You may delete these instructions as you begin to type your report here. \\[2ex]
|
||||||
|
\textbf{Follow us @COMAPMath on X or COMAPCHINAOFFICIAL on Weibo for the \newline
|
||||||
|
most up to date contest information.}
|
||||||
|
}
|
||||||
|
\end{center}
|
||||||
|
% to here
|
||||||
|
%%%%%%%%%%% End Summary %%%%%%%%%%%
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\clearpage
|
||||||
|
\pagestyle{fancy}
|
||||||
|
% Uncomment the next line to generate a Table of Contents
|
||||||
|
%\tableofcontents
|
||||||
|
\newpage
|
||||||
|
\setcounter{page}{1}
|
||||||
|
\rhead{Page \thepage\ }
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
Begin your paper here
|
||||||
|
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\end{document}
|
||||||
|
\end
|
||||||
@@ -0,0 +1,371 @@
|
|||||||
|
[
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "2026 MCM",
|
||||||
|
"text_level": 1,
|
||||||
|
"bbox": [
|
||||||
|
447,
|
||||||
|
90,
|
||||||
|
547,
|
||||||
|
107
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Problem A: Modeling Smartphone Battery Drain",
|
||||||
|
"text_level": 1,
|
||||||
|
"bbox": [
|
||||||
|
290,
|
||||||
|
109,
|
||||||
|
705,
|
||||||
|
128
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "image",
|
||||||
|
"img_path": "images/2b5d356fc542fea6f13c92bb8588b00f0a8255033472c6e0203c1fd673ff1541.jpg",
|
||||||
|
"image_caption": [
|
||||||
|
"25%"
|
||||||
|
],
|
||||||
|
"image_footnote": [],
|
||||||
|
"bbox": [
|
||||||
|
336,
|
||||||
|
162,
|
||||||
|
406,
|
||||||
|
261
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "image",
|
||||||
|
"img_path": "images/9ec42ac02eb37acd1f63743c114bad0fcbcb739f6e35dde2e852dcc98423bdd9.jpg",
|
||||||
|
"image_caption": [
|
||||||
|
"50%"
|
||||||
|
],
|
||||||
|
"image_footnote": [],
|
||||||
|
"bbox": [
|
||||||
|
419,
|
||||||
|
162,
|
||||||
|
488,
|
||||||
|
261
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "image",
|
||||||
|
"img_path": "images/c3baaf0ddbb7528ec969bc9227d45634e34eff9369df76283e9126574c54565e.jpg",
|
||||||
|
"image_caption": [
|
||||||
|
"75%"
|
||||||
|
],
|
||||||
|
"image_footnote": [],
|
||||||
|
"bbox": [
|
||||||
|
501,
|
||||||
|
162,
|
||||||
|
571,
|
||||||
|
261
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "image",
|
||||||
|
"img_path": "images/7c3765aec34f9e2c3dd62910497dfdf5184b7ed353bc82155c9d5938a421a03d.jpg",
|
||||||
|
"image_caption": [
|
||||||
|
"100%"
|
||||||
|
],
|
||||||
|
"image_footnote": [],
|
||||||
|
"bbox": [
|
||||||
|
584,
|
||||||
|
162,
|
||||||
|
656,
|
||||||
|
261
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Smartphones are indispensable tools in modern life, yet their battery behavior often seems unpredictable. On some days a phone may last the whole day; on other days it drains rapidly before lunch. Although some users attribute this to \"heavy use,\" the true drivers of battery depletion are more complex. Power consumption depends on the interplay of screen size and brightness, processor load, network activity, and background applications that continue drawing energy even when the device appears idle. Environmental conditions such as temperature further complicate matters: some batteries lose effective capacity in cold weather and may overheat under sustained heavy use. A battery's behavior is also influenced by its history and how it has been charged during its lifetime.",
|
||||||
|
"bbox": [
|
||||||
|
109,
|
||||||
|
321,
|
||||||
|
880,
|
||||||
|
501
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Your task is to develop a continuous-time mathematical model of a smartphone's battery that returns the state of charge (SOC) as a function of time under realistic usage conditions. This will be used to predict the remaining time-to-empty under different conditions. You should assume that the phone has a lithium-ion battery.",
|
||||||
|
"bbox": [
|
||||||
|
109,
|
||||||
|
508,
|
||||||
|
859,
|
||||||
|
589
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Requirements:",
|
||||||
|
"text_level": 1,
|
||||||
|
"bbox": [
|
||||||
|
112,
|
||||||
|
617,
|
||||||
|
240,
|
||||||
|
635
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "1. Continuous-Time Model: Develop a model to represent the state of charge using a continuous-time equation or system of equations. You may want to begin with the simplest reasonable description of battery drain and then extend it to incorporate additional contributors such as screen usage, processor load, network connections, GPS usage, and other background tasks.",
|
||||||
|
"bbox": [
|
||||||
|
142,
|
||||||
|
643,
|
||||||
|
866,
|
||||||
|
743
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Data as support, not substitute: You may collect or use data for parameter estimation and validation. If open datasets are limited, you may use published measurements or specifications (with proper citation), provided parameters are clearly justified and validated for plausibility. However, projects based solely on discrete curve fitting, timestep regression, or black-box machine learning without an explicit continuous-time model will not satisfy this problem's requirements. All data used must be well documented and freely available, and the data must be free for use under an open license.",
|
||||||
|
"bbox": [
|
||||||
|
169,
|
||||||
|
765,
|
||||||
|
877,
|
||||||
|
904
|
||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "footer",
|
||||||
|
"text": "©2026 by COMAP | www.mathmodels.org | www.mathmodels.org | info@comap.org |",
|
||||||
|
"bbox": [
|
||||||
|
223,
|
||||||
|
938,
|
||||||
|
769,
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|
955
|
||||||
|
],
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||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "2. Time-to-Empty predictions: Use your model to compute or approximate the time-to-empty under various initial charge levels and usage scenarios. Compare predictions to observed or plausible behavior, quantify uncertainty, and identify where the model performs well or poorly.",
|
||||||
|
"bbox": [
|
||||||
|
140,
|
||||||
|
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|
"page_idx": 1
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},
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||||||
|
{
|
||||||
|
"type": "list",
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|
|||||||
|
# 2026 MCM
|
||||||
|
|
||||||
|
# Problem A: Modeling Smartphone Battery Drain
|
||||||
|
|
||||||
|
|
||||||
|
25%
|
||||||
|
|
||||||
|
|
||||||
|
50%
|
||||||
|
|
||||||
|
|
||||||
|
75%
|
||||||
|
|
||||||
|
|
||||||
|
100%
|
||||||
|
|
||||||
|
Smartphones are indispensable tools in modern life, yet their battery behavior often seems unpredictable. On some days a phone may last the whole day; on other days it drains rapidly before lunch. Although some users attribute this to "heavy use," the true drivers of battery depletion are more complex. Power consumption depends on the interplay of screen size and brightness, processor load, network activity, and background applications that continue drawing energy even when the device appears idle. Environmental conditions such as temperature further complicate matters: some batteries lose effective capacity in cold weather and may overheat under sustained heavy use. A battery's behavior is also influenced by its history and how it has been charged during its lifetime.
|
||||||
|
|
||||||
|
Your task is to develop a continuous-time mathematical model of a smartphone's battery that returns the state of charge (SOC) as a function of time under realistic usage conditions. This will be used to predict the remaining time-to-empty under different conditions. You should assume that the phone has a lithium-ion battery.
|
||||||
|
|
||||||
|
# Requirements:
|
||||||
|
|
||||||
|
1. Continuous-Time Model: Develop a model to represent the state of charge using a continuous-time equation or system of equations. You may want to begin with the simplest reasonable description of battery drain and then extend it to incorporate additional contributors such as screen usage, processor load, network connections, GPS usage, and other background tasks.
|
||||||
|
|
||||||
|
Data as support, not substitute: You may collect or use data for parameter estimation and validation. If open datasets are limited, you may use published measurements or specifications (with proper citation), provided parameters are clearly justified and validated for plausibility. However, projects based solely on discrete curve fitting, timestep regression, or black-box machine learning without an explicit continuous-time model will not satisfy this problem's requirements. All data used must be well documented and freely available, and the data must be free for use under an open license.
|
||||||
|
|
||||||
|
2. Time-to-Empty predictions: Use your model to compute or approximate the time-to-empty under various initial charge levels and usage scenarios. Compare predictions to observed or plausible behavior, quantify uncertainty, and identify where the model performs well or poorly.
|
||||||
|
|
||||||
|
○ Show how your model explains differences in these outcomes and identify the specific drivers of rapid battery drain in each case.
|
||||||
|
- Which activities or conditions produce the greatest reductions in battery life? Which ones change the model surprisingly little?
|
||||||
|
|
||||||
|
3. Sensitivity and Assumptions: Examine how your predictions vary after making changes in your modeling assumptions, parameter values, and fluctuations in usage patterns.
|
||||||
|
|
||||||
|
4. Recommendations: Translate your findings into practical recommendations for a cellphone user. For example, which user behaviors—such as reducing brightness, disabling background tasks, or switching network modes—yield the largest improvements in battery life? How might an operating system implement more effective power-saving strategies based on insights from your model? Consider how battery aging reduces effective capacity or how your modeling framework could generalize to other portable devices.
|
||||||
|
|
||||||
|
# Your report should present:
|
||||||
|
|
||||||
|
- A clear description of your model and governing equations.
|
||||||
|
- The assumptions and rationale behind your design choices.
|
||||||
|
Parameter estimation methods and validation results.
|
||||||
|
- A discussion of strengths, limitations, and possible extensions.
|
||||||
|
- An executive-style summary highlighting main results, insights, and recommendations.
|
||||||
|
|
||||||
|
Important: Your model must be grounded in clearly defined physical or mechanical reasoning; discrete curve fitting or other mathematical forms that are disconnected from an explicit continuous-time description of battery behavior will not satisfy the requirements. Projects that rely solely on discrete curve fitting or statistical regression without a clearly formulated continuous-time model will not satisfy the requirements of this problem.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
In-text Citations and A Reference List.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
Smartphone: is a mobile device that combines the functionality of a traditional cell phone with advanced computing capabilities.
|
||||||
|
|
||||||
|
Power Consumption: the rate at which a device uses electrical energy from its battery or power source.
|
||||||
|
|
||||||
|
Processor Load: the actual amount of work being done by the processor at a given moment.
|
||||||
|
|
||||||
|
State of Charge (SOC): a measure of how much energy remains in a battery compared to its full capacity, expressed as a percentage.
|
||||||
|
|
||||||
|
Time-to-Empty: the estimated amount of time remaining before a battery is completely discharged.
|
||||||
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"text": "Imagine a future where it's possible for anyone to visit space by taking a leisurely and scenic ride from the Equator to Earth's orbit and then catching a routine, safe, and inexpensive rocket flight to the Moon, Mars, or beyond. In this future, we could build lush, green, and beautiful space habitats with artificial gravity, where people would vacation, work, or even live. These habitats would alleviate pressure on Earth's delicate, overworked, and fragile ecosystems. The technology to enable these events would provide humankind with limitless, safe, routine, environmentally friendly, efficient, and global access to space. To achieve these goals, some people envision a Space Elevator System, powered by electricity, offering a scalable infrastructure for interplanetary logistics, commerce, and exploration.",
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"text": "At its final operating configuration, the Space Elevator System would comprise three Galactic Harbours, ideally separated by 120 degrees around the equator. Each Galactic Harbour would include a single Earth port with two $100,000\\mathrm{km}$ -long tethers connected to two apex anchors, with multiple space elevators operating together, each capable of lifting massive payloads daily from Earth to geosynchronous orbit (GEO) and beyond to the apex anchor where they can be loaded on a rocket and delivered anywhere using much less fuel.",
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"text": "The Moon Colony Management (MCM) Agency is preparing to build a Moon Colony with an estimated 100,000 people beginning in the year 2050, after completion of the Space Elevator System. It is estimated that the Moon Colony will need about 100 million metric tons of materials. Additionally, water and supplies will routinely need to be sent to sustain the Moon's population once the colony is complete. To get to the Moon, the Galactic Harbour must send material in two steps: first, from the Earth port to the apex anchor via a space elevator, and second, from the apex anchor to the Moon Colony via a rocket. The MCM Agency anticipates that the Galactic Harbour will provide an advanced lift system capable of moving 179,000 metric tons every year, while generating no atmospheric pollution.",
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"text": "The agency is also considering using traditional rockets to supply material for construction and supplies to the Moon Colony. The Earth current has ten rocket launch sites: Alaska, California, Texas, Florida, and Virginia (United States), Kazakhstan, French Guiana, Satish Dhawan Space Centre (India), Taiyuan Satellite Launch Center (China), and Mahia Peninsula (New Zealand).",
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"text": "A rocket would require a single step from the rocket launch site on Earth to the Moon Colony. By 2050 it is estimated that rockets will be able to carry 100-150 metric tons of payload to the Moon using advanced Falcon Heavy launches. You may assume perfect conditions for both the Galactic Harbour system (e.g., no swaying of the tether) and rocket launches (e.g., no failed launches). You should consider the cost and timeline to deliver the materials from the surface of the Earth to the Moon Colony site for the different scenarios.",
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"text": "Your Task:",
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||||||
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"text": "Your task is to utilize a mathematical model to determine the cost and associated timeline in order to transport material to build a 100,000 person Moon Colony starting in 2050. You will need to compare the Modern-Day Space Elevator System's three Galactic Harbours to traditional rockets launched from selected rocket bases.",
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|
||||||
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|
||||||
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"text": "Your model should include:",
|
||||||
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"text_level": 1,
|
||||||
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|
||||||
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||||||
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"text": "1. Consideration of three different scenarios for how the 100 million metric tons of materials will be delivered to build the 100,000-person Moon Colony;",
|
||||||
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|
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|
||||||
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|
||||||
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|
||||||
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"list_items": [
|
||||||
|
"a. using the Space Elevator System's three Galactic Harbor's alone,",
|
||||||
|
"b. traditional rocket launches from existing bases alone (you may choose which facilities to use), or,",
|
||||||
|
"c. some combination of the two methods."
|
||||||
|
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|
||||||
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|
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|
||||||
|
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|
||||||
|
"list_items": [
|
||||||
|
"2. To what extent does your solution(s) change if the transportation systems are not in perfect working order (e.g, swaying of the tether, rockets fail, elevators break, etc.).?",
|
||||||
|
"3. Investigate the water needs for a one-year period once the 100,000-person Moon Colony is fully operational. Use your delivery model to understand the additional cost and timeline needed to ensure the colony has sufficient water for one full year after the Moon Colony is inhabited.",
|
||||||
|
"4. Discuss the impact on the Earth's environment for achieving the 100,000-person Moon Colony under the different scenarios. How would you adjust your model to minimize the environmental impact?",
|
||||||
|
"5. Write a one-page letter recommending a course of action to the fictional MCM Agency to build and sustain a 100,000-person Moon Colony."
|
||||||
|
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|
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|
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|
||||||
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|
||||||
|
"type": "text",
|
||||||
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"text": "Your PDF solution of no more than 25 total pages should include:",
|
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|
||||||
|
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|
||||||
|
"- Your complete solution.",
|
||||||
|
"One-page letter to MCM Agency",
|
||||||
|
"References list.",
|
||||||
|
"AI Use Report (If used does not count toward the 25-page limit.)"
|
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|
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|
"type": "text",
|
||||||
|
"text": "Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.",
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|
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|
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|
||||||
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"type": "text",
|
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"text": "Glossary",
|
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|
"text_level": 1,
|
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|
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|
||||||
|
"type": "text",
|
||||||
|
"text": "Space Elevator System is comprised of three Galactic Harbours plus additional support facilities.",
|
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|
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"text": "Galactic Harbour is comprised of two apex anchors each connected by two tethers to a single Earth Port.",
|
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|
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|
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|
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|
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|
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"text": "Earth Port is the location on the Earth that provides surface support for the Galactic Harbour.",
|
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|
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|
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|
||||||
|
"type": "text",
|
||||||
|
"text": "Tethers are $100,000\\mathrm{km}$ long graphene material that links the Earth port and apex anchors in the Space Elevator System.",
|
||||||
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|
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|
||||||
|
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|
||||||
|
"text": "Apex Anchor is the counterweight in space at the end of the $100,000\\mathrm{km}$ tether.",
|
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|
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|
"text": "Geosynchronous orbit (GEO) is approximately $35,786\\mathrm{km}$ above the surface of the Earth where the orbital period to circle Earth is 24 hours, matching Earth's rotation so it stays over the same longitude each day.",
|
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|
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"text": "Moon Colony is a habitat on the moon with the capacity to support 100,000-persons.",
|
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"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org |",
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|
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]
|
||||||
@@ -0,0 +1,459 @@
|
|||||||
|
[
|
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|
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|
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|
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"content": "At its final operating configuration, the Space Elevator System would comprise three Galactic Harbours, ideally separated by 120 degrees around the equator. Each Galactic Harbour would include a single Earth port with two \\(100,000\\mathrm{km}\\)-long tethers connected to two apex anchors, with multiple space elevators operating together, each capable of lifting massive payloads daily from Earth to geosynchronous orbit (GEO) and beyond to the apex anchor where they can be loaded on a rocket and delivered anywhere using much less fuel."
|
||||||
|
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|
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"content": "The Moon Colony Management (MCM) Agency is preparing to build a Moon Colony with an estimated 100,000 people beginning in the year 2050, after completion of the Space Elevator System. It is estimated that the Moon Colony will need about 100 million metric tons of materials. Additionally, water and supplies will routinely need to be sent to sustain the Moon's population once the colony is complete. To get to the Moon, the Galactic Harbour must send material in two steps: first, from the Earth port to the apex anchor via a space elevator, and second, from the apex anchor to the Moon Colony via a rocket. The MCM Agency anticipates that the Galactic Harbour will provide an advanced lift system capable of moving 179,000 metric tons every year, while generating no atmospheric pollution."
|
||||||
|
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|
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|
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"content": "A rocket would require a single step from the rocket launch site on Earth to the Moon Colony. By 2050 it is estimated that rockets will be able to carry 100-150 metric tons of payload to the Moon using advanced Falcon Heavy launches. You may assume perfect conditions for both the Galactic Harbour system (e.g., no swaying of the tether) and rocket launches (e.g., no failed launches). You should consider the cost and timeline to deliver the materials from the surface of the Earth to the Moon Colony site for the different scenarios."
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"content": "1. Consideration of three different scenarios for how the 100 million metric tons of materials will be delivered to build the 100,000-person Moon Colony;"
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"content": "a. using the Space Elevator System's three Galactic Harbor's alone,"
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"content": "b. traditional rocket launches from existing bases alone (you may choose which facilities to use), or,"
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"content": "c. some combination of the two methods."
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"content": "2. To what extent does your solution(s) change if the transportation systems are not in perfect working order (e.g, swaying of the tether, rockets fail, elevators break, etc.).?"
|
||||||
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"content": "3. Investigate the water needs for a one-year period once the 100,000-person Moon Colony is fully operational. Use your delivery model to understand the additional cost and timeline needed to ensure the colony has sufficient water for one full year after the Moon Colony is inhabited."
|
||||||
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"angle": 0,
|
||||||
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"content": "4. Discuss the impact on the Earth's environment for achieving the 100,000-person Moon Colony under the different scenarios. How would you adjust your model to minimize the environmental impact?"
|
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|
},
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"angle": 0,
|
||||||
|
"content": "5. Write a one-page letter recommending a course of action to the fictional MCM Agency to build and sustain a 100,000-person Moon Colony."
|
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|
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],
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"angle": 0,
|
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|
"content": "Your PDF solution of no more than 25 total pages should include:"
|
||||||
|
},
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{
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"angle": 0,
|
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"content": "One-page Summary Sheet."
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},
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"angle": 0,
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"content": "Table of Contents."
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},
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"angle": 0,
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"content": "- Your complete solution."
|
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},
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"angle": 0,
|
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"content": "One-page letter to MCM Agency"
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"angle": 0,
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"content": "References list."
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"angle": 0,
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"content": "AI Use Report (If used does not count toward the 25-page limit.)"
|
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"angle": 0,
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"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org"
|
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|
}
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],
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[
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"angle": 0,
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"content": "Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution."
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},
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"angle": 0,
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"content": "Glossary"
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},
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"angle": 0,
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"content": "Space Elevator System is comprised of three Galactic Harbours plus additional support facilities."
|
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},
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],
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"angle": 0,
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"content": "Galactic Harbour is comprised of two apex anchors each connected by two tethers to a single Earth Port."
|
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},
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{
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"type": "text",
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"angle": 0,
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"content": "Earth Port is the location on the Earth that provides surface support for the Galactic Harbour."
|
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},
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"type": "text",
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],
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"angle": 0,
|
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"content": "Tethers are \\(100,000\\mathrm{km}\\) long graphene material that links the Earth port and apex anchors in the Space Elevator System."
|
||||||
|
},
|
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|
{
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],
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"angle": 0,
|
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|
"content": "Apex Anchor is the counterweight in space at the end of the \\(100,000\\mathrm{km}\\) tether."
|
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|
},
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{
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],
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"angle": 0,
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"content": "Geosynchronous orbit (GEO) is approximately \\(35,786\\mathrm{km}\\) above the surface of the Earth where the orbital period to circle Earth is 24 hours, matching Earth's rotation so it stays over the same longitude each day."
|
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|
},
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],
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"angle": 0,
|
||||||
|
"content": "Moon Colony is a habitat on the moon with the capacity to support 100,000-persons."
|
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|
},
|
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|
{
|
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"type": "footer",
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"angle": 0,
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"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org |"
|
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|
}
|
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|
]
|
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|
]
|
||||||
Binary file not shown.
@@ -0,0 +1,59 @@
|
|||||||
|
# 2026 MCM
|
||||||
|
|
||||||
|
# Problem B: Creating a Moon Colony Using a Space Elevator System
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
Imagine a future where it's possible for anyone to visit space by taking a leisurely and scenic ride from the Equator to Earth's orbit and then catching a routine, safe, and inexpensive rocket flight to the Moon, Mars, or beyond. In this future, we could build lush, green, and beautiful space habitats with artificial gravity, where people would vacation, work, or even live. These habitats would alleviate pressure on Earth's delicate, overworked, and fragile ecosystems. The technology to enable these events would provide humankind with limitless, safe, routine, environmentally friendly, efficient, and global access to space. To achieve these goals, some people envision a Space Elevator System, powered by electricity, offering a scalable infrastructure for interplanetary logistics, commerce, and exploration.
|
||||||
|
|
||||||
|
At its final operating configuration, the Space Elevator System would comprise three Galactic Harbours, ideally separated by 120 degrees around the equator. Each Galactic Harbour would include a single Earth port with two $100,000\mathrm{km}$ -long tethers connected to two apex anchors, with multiple space elevators operating together, each capable of lifting massive payloads daily from Earth to geosynchronous orbit (GEO) and beyond to the apex anchor where they can be loaded on a rocket and delivered anywhere using much less fuel.
|
||||||
|
|
||||||
|
The Moon Colony Management (MCM) Agency is preparing to build a Moon Colony with an estimated 100,000 people beginning in the year 2050, after completion of the Space Elevator System. It is estimated that the Moon Colony will need about 100 million metric tons of materials. Additionally, water and supplies will routinely need to be sent to sustain the Moon's population once the colony is complete. To get to the Moon, the Galactic Harbour must send material in two steps: first, from the Earth port to the apex anchor via a space elevator, and second, from the apex anchor to the Moon Colony via a rocket. The MCM Agency anticipates that the Galactic Harbour will provide an advanced lift system capable of moving 179,000 metric tons every year, while generating no atmospheric pollution.
|
||||||
|
|
||||||
|
The agency is also considering using traditional rockets to supply material for construction and supplies to the Moon Colony. The Earth current has ten rocket launch sites: Alaska, California, Texas, Florida, and Virginia (United States), Kazakhstan, French Guiana, Satish Dhawan Space Centre (India), Taiyuan Satellite Launch Center (China), and Mahia Peninsula (New Zealand).
|
||||||
|
|
||||||
|
A rocket would require a single step from the rocket launch site on Earth to the Moon Colony. By 2050 it is estimated that rockets will be able to carry 100-150 metric tons of payload to the Moon using advanced Falcon Heavy launches. You may assume perfect conditions for both the Galactic Harbour system (e.g., no swaying of the tether) and rocket launches (e.g., no failed launches). You should consider the cost and timeline to deliver the materials from the surface of the Earth to the Moon Colony site for the different scenarios.
|
||||||
|
|
||||||
|
# Your Task:
|
||||||
|
|
||||||
|
Your task is to utilize a mathematical model to determine the cost and associated timeline in order to transport material to build a 100,000 person Moon Colony starting in 2050. You will need to compare the Modern-Day Space Elevator System's three Galactic Harbours to traditional rockets launched from selected rocket bases.
|
||||||
|
|
||||||
|
# Your model should include:
|
||||||
|
|
||||||
|
1. Consideration of three different scenarios for how the 100 million metric tons of materials will be delivered to build the 100,000-person Moon Colony;
|
||||||
|
|
||||||
|
a. using the Space Elevator System's three Galactic Harbor's alone,
|
||||||
|
b. traditional rocket launches from existing bases alone (you may choose which facilities to use), or,
|
||||||
|
c. some combination of the two methods.
|
||||||
|
|
||||||
|
2. To what extent does your solution(s) change if the transportation systems are not in perfect working order (e.g, swaying of the tether, rockets fail, elevators break, etc.).?
|
||||||
|
3. Investigate the water needs for a one-year period once the 100,000-person Moon Colony is fully operational. Use your delivery model to understand the additional cost and timeline needed to ensure the colony has sufficient water for one full year after the Moon Colony is inhabited.
|
||||||
|
4. Discuss the impact on the Earth's environment for achieving the 100,000-person Moon Colony under the different scenarios. How would you adjust your model to minimize the environmental impact?
|
||||||
|
5. Write a one-page letter recommending a course of action to the fictional MCM Agency to build and sustain a 100,000-person Moon Colony.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One-page letter to MCM Agency
|
||||||
|
References list.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
Space Elevator System is comprised of three Galactic Harbours plus additional support facilities.
|
||||||
|
|
||||||
|
Galactic Harbour is comprised of two apex anchors each connected by two tethers to a single Earth Port.
|
||||||
|
|
||||||
|
Earth Port is the location on the Earth that provides surface support for the Galactic Harbour.
|
||||||
|
|
||||||
|
Tethers are $100,000\mathrm{km}$ long graphene material that links the Earth port and apex anchors in the Space Elevator System.
|
||||||
|
|
||||||
|
Apex Anchor is the counterweight in space at the end of the $100,000\mathrm{km}$ tether.
|
||||||
|
|
||||||
|
Geosynchronous orbit (GEO) is approximately $35,786\mathrm{km}$ above the surface of the Earth where the orbital period to circle Earth is 24 hours, matching Earth's rotation so it stays over the same longitude each day.
|
||||||
|
|
||||||
|
Moon Colony is a habitat on the moon with the capacity to support 100,000-persons.
|
||||||
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|
[
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "2026 MCM Problem C: Data With The Stars",
|
||||||
|
"text_level": 1,
|
||||||
|
"bbox": [
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||||||
|
],
|
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|
"page_idx": 0
|
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},
|
||||||
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{
|
||||||
|
"type": "image",
|
||||||
|
"img_path": "images/4bc01e42be38c629fa388c4fe8c64a650dc2f0561d90ac656f9e789bcebaf7aa.jpg",
|
||||||
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"image_caption": [],
|
||||||
|
"image_footnote": [],
|
||||||
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"bbox": [
|
||||||
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|
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|
313
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||||||
|
],
|
||||||
|
"page_idx": 0
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Dancing with the Stars (DWTS) is the American version of an international television franchise based on the British show \"Strictly Come Dancing\" (\"Come Dancing\" originally). Versions of the show have appeared in Albania, Argentina, Australia, China, France, India, and many other countries. The U.S. version, the focus of this problem, has completed 34 seasons.",
|
||||||
|
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|
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],
|
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"page_idx": 0
|
||||||
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},
|
||||||
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{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Celebrities are partnered with professional dancers and then perform dances each week. A panel of expert judges scores each couple's dance, and fans vote (by phone or online) for their favorite couple that week. Fans can vote once or multiple times up to a limit announced each week. Further, fans vote for the star they wish to keep, but cannot vote to eliminate a star. The judge and fan votes are combined in order to determine which couple to eliminate (the lowest combined score) that week. Three (in some seasons more) couples reach the finals and in the week of the finals the combined scores from fans and judges are used to rank them from $1^{\\text{st}}$ to $3^{\\text{rd}}$ (or $4^{\\text{th}}$ , $5^{\\text{th}}$ ).",
|
||||||
|
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|
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|
||||||
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},
|
||||||
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|
||||||
|
"type": "text",
|
||||||
|
"text": "There are many possible methods of combining fan votes and judge scores. In the first two seasons of the U.S. show, the combination was based on ranks. Season 2 concerns (due to celebrity contestant Jerry Rice who was a finalist despite very low judge scores) led to a modification to use percentages instead of ranks. Examples of these two approaches are provided in the Appendix.",
|
||||||
|
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|
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|
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},
|
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|
||||||
|
"type": "text",
|
||||||
|
"text": "In season 27, another \"controversy\" occurred when celebrity contestant Bobby Bones won despite consistently low judges scores. In response, starting in season 28 a slight modification to the elimination process was made. The bottom two contestants were identified using the combined judge scores and fan votes, and then during the live show the judges voted to select which of these two to eliminate. Around this same season, the producers also returned to using the method of ranks to combine judges scores with fan votes as in seasons one and two. The exact season this change occurred is not known, but it is reasonable to assume it was season 28.",
|
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|
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||||||
|
"type": "text",
|
||||||
|
"text": "Judge scores are meant to reflect which dancers are technically better, although there is some subjectivity in what makes a dance better. Fan votes are likely much more subjective, influenced by the quality of the dance, but also the popularity and charisma of the celebrity. Show producers might actually prefer, to some extent, conflicts in opinions and votes as such occurrences boost fan interest and excitement.",
|
||||||
|
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|
||||||
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},
|
||||||
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{
|
||||||
|
"type": "footer",
|
||||||
|
"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org",
|
||||||
|
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|
||||||
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},
|
||||||
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{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Data with judges scores and contestant information is provided and described below. You may choose to include additional information or other data at your discretion, but you must completely document the sources. Use the data to:",
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
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|
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"page_idx": 1
|
||||||
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},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "- Develop a mathematical model (or models) to produce estimated fan votes (which are unknown and a closely guarded secret) for each contestant for the weeks they competed.",
|
||||||
|
"bbox": [
|
||||||
|
111,
|
||||||
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|
||||||
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|
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],
|
||||||
|
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||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
"sub_type": "text",
|
||||||
|
"list_items": [
|
||||||
|
"- Does your model correctly estimate fan votes that lead to results consistent with who was eliminated each week? Provide measures of the consistency.",
|
||||||
|
"- How much certainty is there in the fan vote totals you produced, and is that certainty always the same for each contestant/week? Provide measures of your certainty for the estimates."
|
||||||
|
],
|
||||||
|
"bbox": [
|
||||||
|
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"page_idx": 1
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "- Use your fan vote estimates with the rest of the data to:",
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
|
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|
||||||
|
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|
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|
"page_idx": 1
|
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|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
"sub_type": "text",
|
||||||
|
"list_items": [
|
||||||
|
"○ Compare and contrast the results produced by the two approaches used by the show to combine judge and fan votes (i.e. rank and percentage) across seasons (i.e. apply both approaches to each season). If differences in outcomes exist, does one method seem to favor fan votes more than the other?",
|
||||||
|
"○ Examine the two voting methods applied to specific celebrities where there was “controversy”, meaning differences between judges and fans. Would the choice of method to combine judge scores and fan votes have led to the same result for each of these contestants? How would including the additional approach of having judges choose which of the bottom two couples to eliminate each week impact the results? Some examples you might consider (there may also be others you identified):"
|
||||||
|
],
|
||||||
|
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|
||||||
|
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|
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
"sub_type": "text",
|
||||||
|
"list_items": [
|
||||||
|
"- season 2 - Jerry Rice, runner up despite the lowest judges scores in 5 weeks.",
|
||||||
|
"- season 4 - Billy Ray Cyrus was $5^{\\text{th}}$ despite last place judge scores in 6 weeks.",
|
||||||
|
"- season 11 - Bristol Palin was $3^{\\text{rd}}$ with the lowest judge scores 12 times.",
|
||||||
|
"- season 27 - Bobby Bones won the despite consistently low judges scores"
|
||||||
|
],
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
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||||||
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563
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],
|
||||||
|
"page_idx": 1
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "- Based on your analysis, which of the two methods would you recommend using for future seasons and why? Would you suggest including the additional approach of judges choosing from the bottom two couples?",
|
||||||
|
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|
||||||
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||||||
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"sub_type": "text",
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||||||
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"list_items": [
|
||||||
|
"- Use the data including your fan vote estimates to develop a model that analyzes the impact of various pro dancers as well as characteristics for the celebrities available in the data (age, industry, etc). How much do such things impact how well a celebrity will do in the competition? Do they impact judges scores and fan votes in the same way?",
|
||||||
|
"- Propose another system using fan votes and judge scores each week that you believe is more \"fair\" (or \"better\" in some other way such as making the show more exciting for the fans). Provide support for why your approach should be adopted by the show producers.",
|
||||||
|
"- Produce a report of no more than 25 pages with your findings and include a one- to two-page memo summarizing your results with advice for producers of DWTS on the impact of how judge and fan votes are combined with recommendations for how to do so in future seasons."
|
||||||
|
],
|
||||||
|
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||||||
|
"page_idx": 1
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||||||
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},
|
||||||
|
{
|
||||||
|
"type": "footer",
|
||||||
|
"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org",
|
||||||
|
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|
||||||
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{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Your PDF solution of no more than 25 total pages should include:",
|
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|
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|
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|
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"page_idx": 2
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{
|
||||||
|
"type": "list",
|
||||||
|
"sub_type": "text",
|
||||||
|
"list_items": [
|
||||||
|
"One-page Summary Sheet.",
|
||||||
|
"Table of Contents.",
|
||||||
|
"- Your complete solution.",
|
||||||
|
"One- to two-page memo.",
|
||||||
|
"- References list.",
|
||||||
|
"AI Use Report (If used does not count toward the 25-page limit.)"
|
||||||
|
],
|
||||||
|
"bbox": [
|
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|
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||||||
|
"page_idx": 2
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.",
|
||||||
|
"bbox": [
|
||||||
|
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|
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|
||||||
|
"page_idx": 2
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Data File: 2026_MCM_Problem_C_Data.csv – contestant information, results, and judges scores by week for seasons 1 – 34. The data description is provided in Table 1.",
|
||||||
|
"bbox": [
|
||||||
|
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|
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|
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|
"page_idx": 2
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "table",
|
||||||
|
"img_path": "images/cb329b2b026f81b7a3a204c8515f17dd53f9540f9b942d238a3168154e85ad5a.jpg",
|
||||||
|
"table_caption": [
|
||||||
|
"Table 1: Data Description for 2026_MCM_Problem_C_Data.csv"
|
||||||
|
],
|
||||||
|
"table_footnote": [],
|
||||||
|
"table_body": "<table><tr><td>Variables</td><td>Explanation</td><td>Example</td></tr><tr><td>celebrity_name</td><td>Name of celebrity contestant (Star)</td><td>Jerry Rice, Mark Cuban, ...</td></tr><tr><td>ballroompartner</td><td>Name of professional dancer partner</td><td>Cheryl Burke, Derek Hough, ...</td></tr><tr><td>celebrity_industry</td><td>Star profession category</td><td>Athlete, Model, ...</td></tr><tr><td>celebrity_homestate</td><td>Star home state (if from U.S.)</td><td>Ohio, Maine, ...</td></tr><tr><td>celebrity_homecountry/region</td><td>Star home country/region</td><td>United States, England, ...</td></tr><tr><td>celebrity_age during season</td><td>Age of the star in the season</td><td>32, 29, ...</td></tr><tr><td>season</td><td>Season of the show</td><td>1, 2, 3, ..., 32</td></tr><tr><td>results</td><td>Season results for the start</td><td>1st Place, Eliminated Week 2, ...</td></tr><tr><td>placement</td><td>Final place for the season (1 best)</td><td>1, 2, 3, ...</td></tr><tr><td>weekXjudgeY_score</td><td>Score from judge Y in week X</td><td>1, 2, 3, ...</td></tr></table>",
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
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|
||||||
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|
||||||
|
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||||||
|
],
|
||||||
|
"page_idx": 2
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "Notes on the data:",
|
||||||
|
"text_level": 1,
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
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||||||
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|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "1. Judges scores for each dance are from 1 (low) to 10 (high).",
|
||||||
|
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|
||||||
|
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|
||||||
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|
||||||
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|
"page_idx": 2
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
"sub_type": "text",
|
||||||
|
"list_items": [
|
||||||
|
"a. In some weeks the score reported includes a decimal (e.g. 8.5) because each celebrity performed more than one dance and the scores from each are averaged.",
|
||||||
|
"b. In some weeks, bonus points were awarded (dance offs etc); they are spread evenly across judge/dance scores.",
|
||||||
|
"c. Team dance scores were averaged with scores for each individual team member."
|
||||||
|
],
|
||||||
|
"bbox": [
|
||||||
|
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|
"page_idx": 2
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"text": "2. Judges are listed in the order they scored dances; thus \"Judge Y\" may not be the same judge from week to week, or season to season.",
|
||||||
|
"bbox": [
|
||||||
|
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|
||||||
|
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|
||||||
|
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|
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||||||
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"3. The number of celebrities is not the same across the seasons, nor is the number of weeks the show ran.",
|
||||||
|
"4. Season 15 was the only season to feature an all-star cast of returning celebrities.",
|
||||||
|
"5. There are occasionally weeks when no celebrity was eliminated, and others where more than one was eliminated.",
|
||||||
|
"6. $N / A$ values occur in the data set for",
|
||||||
|
"a. the $4^{th}$ judge score if there is not $4^{th}$ judge for that week (usually there are 3) and \nb. in weeks that the show did not run in a season (for example, season 1 lasted 6 weeks so $N / A$ values are recorded for weeks 7 thru 11).",
|
||||||
|
"7. A 0 score is recorded for celebrities who are eliminated. For example, in Season 1 the first celebrity eliminated was Trista Sutter at the end of the Week 2 show. She thus has scores of 0 for the rest of the season (week 3 through week 6)."
|
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"type": "text",
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"text": "Appendix: Examples of Voting Schemes",
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"type": "text",
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"text": "1. COMBINED BY RANK (used in seasons 1, 2, and $28^{\\mathrm{a}}$ - 34)",
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"text": "In seasons 1 and 2 judges and fan votes were combined by rank. For example, in season 1, week 4 there were four remaining contestants. Rachel Hunter was eliminated meaning she received the lowest combined rank. In Table 2 the judges scores and ranks are shown, and we created one possible set of fan votes that would produce the correct result. There are many possible values for fan votes that would also give the same results. You should not use these as actual values as this is just one example. Since Rachel was ranked $2^{\\text{nd}}$ by judges, in order to finish with the lowest combined score, she has the lowest fan vote ( $4^{\\text{th}}$ place) for a total rank of 6.",
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"img_path": "images/a3cf6c88e3f32375d8a0c5f9d0dccfda4983ce98b72a8aa778fc09ca4aa5bbd3.jpg",
|
||||||
|
"table_caption": [
|
||||||
|
"Table 2: Example of Combining Judge and Fan Votes by Rank (Season 1, Week 4)"
|
||||||
|
],
|
||||||
|
"table_footnote": [
|
||||||
|
"* Fan vote/rank are unknown, hypothetical values chosen to produce the correct final ranks"
|
||||||
|
],
|
||||||
|
"table_body": "<table><tr><td>Contestant</td><td>Total Judges Score</td><td>Judges Score Rank</td><td>Fan Vote*</td><td>Fan Rank*</td><td>Sum of ranks</td></tr><tr><td>Rachel Hunter</td><td>25</td><td>2</td><td>1.1 million</td><td>4</td><td>6</td></tr><tr><td>Joey McIntyre</td><td>20</td><td>4</td><td>3.7 million</td><td>1</td><td>5</td></tr><tr><td>John O’Hurley</td><td>21</td><td>3</td><td>3.2 million</td><td>2</td><td>5</td></tr><tr><td>Kelly Monaco</td><td>26</td><td>1</td><td>2 million</td><td>3</td><td>4</td></tr></table>",
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"type": "text",
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"text": "2. COMBINED BY PERCENT (used for season 3 through $27^{\\mathrm{a}}$ )",
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|
"type": "text",
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||||||
|
"text": "Starting in season 3 scores were combined using percents instead of ranks. An example is shown using week 9 of season 5. In that week, Jennie Garth was eliminated. Again, we artificially created fan votes that produce total percents to correctly lead to that result. The judges' percent is computed by dividing the total judge score for the contestant by the sum of total judge scores for all 4 contestants. Based on the judges' percent, Jennie was $3^{\\text{rd}}$ . However, adding the percent of the 10 million artificially created fan votes we assigned to the judges' percent she was $4^{\\text{th}}$ .",
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|
"type": "footer",
|
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|
"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org",
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|
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|
"type": "table",
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|
||||||
|
"table_caption": [
|
||||||
|
"Table 3: Example of Combining Judge and Fan Votes by Percent (Season 5, Week 9)"
|
||||||
|
],
|
||||||
|
"table_footnote": [
|
||||||
|
"* Fan vote is unknown, values hypothetical to produce the correct final standings"
|
||||||
|
],
|
||||||
|
"table_body": "<table><tr><td>Contestant</td><td>Total Judges Score</td><td>Judges Score Percent</td><td>Fan Vote*</td><td>Fan Percent*</td><td>Sum of Percent</td></tr><tr><td>Jennie Garth</td><td>29</td><td>29/117 = 24.8%</td><td>1.1 million</td><td>1.1/10 = 11%</td><td>35.8</td></tr><tr><td>Marie Osmond</td><td>28</td><td>28/117 = 23.9%</td><td>3.7 million</td><td>3.7/10 = 37%</td><td>60.9</td></tr><tr><td>Mel B</td><td>30</td><td>30/117 = 25.6%</td><td>3.2 million</td><td>3.2/10 = 32%</td><td>57.8</td></tr><tr><td>Helio Castroneves</td><td>30</td><td>30/117 = 25.6%</td><td>2 million</td><td>2/10 = 20%</td><td>45.6</td></tr><tr><td>Total</td><td>117</td><td></td><td>10 million</td><td></td><td></td></tr></table>",
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"text": "a The year of the return to the rank based method is not known for certain; season 28 is a reasonable assumption.",
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"text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org",
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@@ -0,0 +1,771 @@
|
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[
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"content": "2026 MCM Problem C: Data With The Stars"
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"content": "Dancing with the Stars (DWTS) is the American version of an international television franchise based on the British show \"Strictly Come Dancing\" (\"Come Dancing\" originally). Versions of the show have appeared in Albania, Argentina, Australia, China, France, India, and many other countries. The U.S. version, the focus of this problem, has completed 34 seasons."
|
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"content": "Celebrities are partnered with professional dancers and then perform dances each week. A panel of expert judges scores each couple's dance, and fans vote (by phone or online) for their favorite couple that week. Fans can vote once or multiple times up to a limit announced each week. Further, fans vote for the star they wish to keep, but cannot vote to eliminate a star. The judge and fan votes are combined in order to determine which couple to eliminate (the lowest combined score) that week. Three (in some seasons more) couples reach the finals and in the week of the finals the combined scores from fans and judges are used to rank them from \\(1^{\\text{st}}\\) to \\(3^{\\text{rd}}\\) (or \\(4^{\\text{th}}\\), \\(5^{\\text{th}}\\))."
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"angle": 0,
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"content": "There are many possible methods of combining fan votes and judge scores. In the first two seasons of the U.S. show, the combination was based on ranks. Season 2 concerns (due to celebrity contestant Jerry Rice who was a finalist despite very low judge scores) led to a modification to use percentages instead of ranks. Examples of these two approaches are provided in the Appendix."
|
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"content": "In season 27, another \"controversy\" occurred when celebrity contestant Bobby Bones won despite consistently low judges scores. In response, starting in season 28 a slight modification to the elimination process was made. The bottom two contestants were identified using the combined judge scores and fan votes, and then during the live show the judges voted to select which of these two to eliminate. Around this same season, the producers also returned to using the method of ranks to combine judges scores with fan votes as in seasons one and two. The exact season this change occurred is not known, but it is reasonable to assume it was season 28."
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"content": "Judge scores are meant to reflect which dancers are technically better, although there is some subjectivity in what makes a dance better. Fan votes are likely much more subjective, influenced by the quality of the dance, but also the popularity and charisma of the celebrity. Show producers might actually prefer, to some extent, conflicts in opinions and votes as such occurrences boost fan interest and excitement."
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"angle": 0,
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"content": "Data with judges scores and contestant information is provided and described below. You may choose to include additional information or other data at your discretion, but you must completely document the sources. Use the data to:"
|
||||||
|
},
|
||||||
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"angle": 0,
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"content": "- Develop a mathematical model (or models) to produce estimated fan votes (which are unknown and a closely guarded secret) for each contestant for the weeks they competed."
|
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},
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],
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"angle": 0,
|
||||||
|
"content": "- Does your model correctly estimate fan votes that lead to results consistent with who was eliminated each week? Provide measures of the consistency."
|
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|
},
|
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{
|
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],
|
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|
"angle": 0,
|
||||||
|
"content": "- How much certainty is there in the fan vote totals you produced, and is that certainty always the same for each contestant/week? Provide measures of your certainty for the estimates."
|
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|
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|
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|
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],
|
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"angle": 0,
|
||||||
|
"content": "- Use your fan vote estimates with the rest of the data to:"
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
|
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|
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"angle": 0,
|
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|
"content": "○ Compare and contrast the results produced by the two approaches used by the show to combine judge and fan votes (i.e. rank and percentage) across seasons (i.e. apply both approaches to each season). If differences in outcomes exist, does one method seem to favor fan votes more than the other?"
|
||||||
|
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|
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],
|
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"angle": 0,
|
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"content": "○ Examine the two voting methods applied to specific celebrities where there was “controversy”, meaning differences between judges and fans. Would the choice of method to combine judge scores and fan votes have led to the same result for each of these contestants? How would including the additional approach of having judges choose which of the bottom two couples to eliminate each week impact the results? Some examples you might consider (there may also be others you identified):"
|
||||||
|
},
|
||||||
|
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|
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||||||
|
"content": "- season 2 - Jerry Rice, runner up despite the lowest judges scores in 5 weeks."
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
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"bbox": [
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],
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"angle": 0,
|
||||||
|
"content": "- season 4 - Billy Ray Cyrus was \\(5^{\\text{th}}\\) despite last place judge scores in 6 weeks."
|
||||||
|
},
|
||||||
|
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|
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|
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],
|
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|
"angle": 0,
|
||||||
|
"content": "- season 11 - Bristol Palin was \\(3^{\\text{rd}}\\) with the lowest judge scores 12 times."
|
||||||
|
},
|
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|
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|
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|
"type": "text",
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],
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|
"angle": 0,
|
||||||
|
"content": "- season 27 - Bobby Bones won the despite consistently low judges scores"
|
||||||
|
},
|
||||||
|
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|
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|
"type": "list",
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"angle": 0,
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"content": null
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|
},
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|
{
|
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|
"type": "text",
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],
|
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|
"angle": 0,
|
||||||
|
"content": "- Based on your analysis, which of the two methods would you recommend using for future seasons and why? Would you suggest including the additional approach of judges choosing from the bottom two couples?"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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"bbox": [
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],
|
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|
"angle": 0,
|
||||||
|
"content": "- Use the data including your fan vote estimates to develop a model that analyzes the impact of various pro dancers as well as characteristics for the celebrities available in the data (age, industry, etc). How much do such things impact how well a celebrity will do in the competition? Do they impact judges scores and fan votes in the same way?"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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"bbox": [
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0.775
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Propose another system using fan votes and judge scores each week that you believe is more \"fair\" (or \"better\" in some other way such as making the show more exciting for the fans). Provide support for why your approach should be adopted by the show producers."
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "text",
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0.846
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Produce a report of no more than 25 pages with your findings and include a one- to two-page memo summarizing your results with advice for producers of DWTS on the impact of how judge and fan votes are combined with recommendations for how to do so in future seasons."
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "list",
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],
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|
"angle": 0,
|
||||||
|
"content": null
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "footer",
|
||||||
|
"bbox": [
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],
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"angle": 0,
|
||||||
|
"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org"
|
||||||
|
}
|
||||||
|
],
|
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|
[
|
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|
{
|
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|
"type": "text",
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],
|
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|
"angle": 0,
|
||||||
|
"content": "Your PDF solution of no more than 25 total pages should include:"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
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|
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "One-page Summary Sheet."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
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|
0.141
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Table of Contents."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "- Your complete solution."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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],
|
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|
"angle": 0,
|
||||||
|
"content": "One- to two-page memo."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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],
|
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|
"angle": 0,
|
||||||
|
"content": "- References list."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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],
|
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|
"angle": 0,
|
||||||
|
"content": "AI Use Report (If used does not count toward the 25-page limit.)"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
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|
"bbox": [
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],
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|
"angle": 0,
|
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|
"content": null
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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"bbox": [
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
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],
|
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|
"angle": 0,
|
||||||
|
"content": "Data File: 2026_MCM_Problem_C_Data.csv – contestant information, results, and judges scores by week for seasons 1 – 34. The data description is provided in Table 1."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "table_caption",
|
||||||
|
"bbox": [
|
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|
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|
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|
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|
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|
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Table 1: Data Description for 2026_MCM_Problem_C_Data.csv"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "table",
|
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|
"bbox": [
|
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|
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|
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|
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|
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|
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "<table><tr><td>Variables</td><td>Explanation</td><td>Example</td></tr><tr><td>celebrity_name</td><td>Name of celebrity contestant (Star)</td><td>Jerry Rice, Mark Cuban, ...</td></tr><tr><td>ballroompartner</td><td>Name of professional dancer partner</td><td>Cheryl Burke, Derek Hough, ...</td></tr><tr><td>celebrity_industry</td><td>Star profession category</td><td>Athlete, Model, ...</td></tr><tr><td>celebrity_homestate</td><td>Star home state (if from U.S.)</td><td>Ohio, Maine, ...</td></tr><tr><td>celebrity_homecountry/region</td><td>Star home country/region</td><td>United States, England, ...</td></tr><tr><td>celebrity_age during season</td><td>Age of the star in the season</td><td>32, 29, ...</td></tr><tr><td>season</td><td>Season of the show</td><td>1, 2, 3, ..., 32</td></tr><tr><td>results</td><td>Season results for the start</td><td>1st Place, Eliminated Week 2, ...</td></tr><tr><td>placement</td><td>Final place for the season (1 best)</td><td>1, 2, 3, ...</td></tr><tr><td>weekXjudgeY_score</td><td>Score from judge Y in week X</td><td>1, 2, 3, ...</td></tr></table>"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "title",
|
||||||
|
"bbox": [
|
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|
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|
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|
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|
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Notes on the data:"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
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|
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|
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|
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|
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|
||||||
|
0.776
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "1. Judges scores for each dance are from 1 (low) to 10 (high)."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
||||||
|
0.17,
|
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|
0.777,
|
||||||
|
0.877,
|
||||||
|
0.81
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "a. In some weeks the score reported includes a decimal (e.g. 8.5) because each celebrity performed more than one dance and the scores from each are averaged."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
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|
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|
0.812,
|
||||||
|
0.858,
|
||||||
|
0.845
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "b. In some weeks, bonus points were awarded (dance offs etc); they are spread evenly across judge/dance scores."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
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|
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|
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|
||||||
|
0.835,
|
||||||
|
0.862
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "c. Team dance scores were averaged with scores for each individual team member."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
||||||
|
"bbox": [
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": null
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
|
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|
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|
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|
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|
0.879,
|
||||||
|
0.899
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "2. Judges are listed in the order they scored dances; thus \"Judge Y\" may not be the same judge from week to week, or season to season."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "footer",
|
||||||
|
"bbox": [
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org"
|
||||||
|
}
|
||||||
|
],
|
||||||
|
[
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
|
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|
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|
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|
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|
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|
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|
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|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "3. The number of celebrities is not the same across the seasons, nor is the number of weeks the show ran."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
||||||
|
"bbox": [
|
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|
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|
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0.782,
|
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|
0.143
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],
|
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|
"angle": 0,
|
||||||
|
"content": "4. Season 15 was the only season to feature an all-star cast of returning celebrities."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
|
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|
0.112,
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|
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0.875,
|
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|
0.178
|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "5. There are occasionally weeks when no celebrity was eliminated, and others where more than one was eliminated."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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|
0.428,
|
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|
0.196
|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "6. \\(N / A\\) values occur in the data set for"
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
|
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|
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|
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|
0.87,
|
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|
0.248
|
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|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "a. the \\(4^{th}\\) judge score if there is not \\(4^{th}\\) judge for that week (usually there are 3) and \nb. in weeks that the show did not run in a season (for example, season 1 lasted 6 weeks so \\(N / A\\) values are recorded for weeks 7 thru 11)."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "text",
|
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|
"bbox": [
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|
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|
||||||
|
],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "7. A 0 score is recorded for celebrities who are eliminated. For example, in Season 1 the first celebrity eliminated was Trista Sutter at the end of the Week 2 show. She thus has scores of 0 for the rest of the season (week 3 through week 6)."
|
||||||
|
},
|
||||||
|
{
|
||||||
|
"type": "list",
|
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|
"bbox": [
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|
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],
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|
"angle": 0,
|
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|
"content": null
|
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|
},
|
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|
{
|
||||||
|
"type": "title",
|
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"angle": 0,
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"content": "Appendix: Examples of Voting Schemes"
|
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},
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"angle": 0,
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"content": "1. COMBINED BY RANK (used in seasons 1, 2, and \\(28^{\\mathrm{a}}\\) - 34)"
|
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},
|
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|
{
|
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],
|
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"angle": 0,
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"content": "In seasons 1 and 2 judges and fan votes were combined by rank. For example, in season 1, week 4 there were four remaining contestants. Rachel Hunter was eliminated meaning she received the lowest combined rank. In Table 2 the judges scores and ranks are shown, and we created one possible set of fan votes that would produce the correct result. There are many possible values for fan votes that would also give the same results. You should not use these as actual values as this is just one example. Since Rachel was ranked \\(2^{\\text{nd}}\\) by judges, in order to finish with the lowest combined score, she has the lowest fan vote (\\(4^{\\text{th}}\\) place) for a total rank of 6."
|
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|
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],
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"angle": 0,
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"content": "Table 2: Example of Combining Judge and Fan Votes by Rank (Season 1, Week 4)"
|
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},
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|
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"type": "table",
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"content": "<table><tr><td>Contestant</td><td>Total Judges Score</td><td>Judges Score Rank</td><td>Fan Vote*</td><td>Fan Rank*</td><td>Sum of ranks</td></tr><tr><td>Rachel Hunter</td><td>25</td><td>2</td><td>1.1 million</td><td>4</td><td>6</td></tr><tr><td>Joey McIntyre</td><td>20</td><td>4</td><td>3.7 million</td><td>1</td><td>5</td></tr><tr><td>John O’Hurley</td><td>21</td><td>3</td><td>3.2 million</td><td>2</td><td>5</td></tr><tr><td>Kelly Monaco</td><td>26</td><td>1</td><td>2 million</td><td>3</td><td>4</td></tr></table>"
|
||||||
|
},
|
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|
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|
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|
"type": "table_footnote",
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],
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"angle": 0,
|
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|
"content": "* Fan vote/rank are unknown, hypothetical values chosen to produce the correct final ranks"
|
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|
},
|
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|
{
|
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|
"type": "title",
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"bbox": [
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],
|
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|
"angle": 0,
|
||||||
|
"content": "2. COMBINED BY PERCENT (used for season 3 through \\(27^{\\mathrm{a}}\\))"
|
||||||
|
},
|
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|
{
|
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|
"type": "text",
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0.81
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],
|
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"angle": 0,
|
||||||
|
"content": "Starting in season 3 scores were combined using percents instead of ranks. An example is shown using week 9 of season 5. In that week, Jennie Garth was eliminated. Again, we artificially created fan votes that produce total percents to correctly lead to that result. The judges' percent is computed by dividing the total judge score for the contestant by the sum of total judge scores for all 4 contestants. Based on the judges' percent, Jennie was \\(3^{\\text{rd}}\\). However, adding the percent of the 10 million artificially created fan votes we assigned to the judges' percent she was \\(4^{\\text{th}}\\)."
|
||||||
|
},
|
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|
{
|
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|
"type": "footer",
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"angle": 0,
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"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org"
|
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|
}
|
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|
],
|
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|
[
|
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{
|
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"type": "table_caption",
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"bbox": [
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|
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],
|
||||||
|
"angle": 0,
|
||||||
|
"content": "Table 3: Example of Combining Judge and Fan Votes by Percent (Season 5, Week 9)"
|
||||||
|
},
|
||||||
|
{
|
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|
"type": "table",
|
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"angle": 0,
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|
"content": "<table><tr><td>Contestant</td><td>Total Judges Score</td><td>Judges Score Percent</td><td>Fan Vote*</td><td>Fan Percent*</td><td>Sum of Percent</td></tr><tr><td>Jennie Garth</td><td>29</td><td>29/117 = 24.8%</td><td>1.1 million</td><td>1.1/10 = 11%</td><td>35.8</td></tr><tr><td>Marie Osmond</td><td>28</td><td>28/117 = 23.9%</td><td>3.7 million</td><td>3.7/10 = 37%</td><td>60.9</td></tr><tr><td>Mel B</td><td>30</td><td>30/117 = 25.6%</td><td>3.2 million</td><td>3.2/10 = 32%</td><td>57.8</td></tr><tr><td>Helio Castroneves</td><td>30</td><td>30/117 = 25.6%</td><td>2 million</td><td>2/10 = 20%</td><td>45.6</td></tr><tr><td>Total</td><td>117</td><td></td><td>10 million</td><td></td><td></td></tr></table>"
|
||||||
|
},
|
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|
{
|
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"type": "table_footnote",
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],
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"angle": 0,
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"content": "* Fan vote is unknown, values hypothetical to produce the correct final standings"
|
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|
},
|
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{
|
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|
"type": "text",
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],
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|
"angle": 0,
|
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|
"content": "a The year of the return to the rank based method is not known for certain; season 28 is a reasonable assumption."
|
||||||
|
},
|
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|
{
|
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"type": "footer",
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"content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org"
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}
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|
]
|
||||||
|
]
|
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|
|||||||
|
# 2026 MCM Problem C: Data With The Stars
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
Dancing with the Stars (DWTS) is the American version of an international television franchise based on the British show "Strictly Come Dancing" ("Come Dancing" originally). Versions of the show have appeared in Albania, Argentina, Australia, China, France, India, and many other countries. The U.S. version, the focus of this problem, has completed 34 seasons.
|
||||||
|
|
||||||
|
Celebrities are partnered with professional dancers and then perform dances each week. A panel of expert judges scores each couple's dance, and fans vote (by phone or online) for their favorite couple that week. Fans can vote once or multiple times up to a limit announced each week. Further, fans vote for the star they wish to keep, but cannot vote to eliminate a star. The judge and fan votes are combined in order to determine which couple to eliminate (the lowest combined score) that week. Three (in some seasons more) couples reach the finals and in the week of the finals the combined scores from fans and judges are used to rank them from $1^{\text{st}}$ to $3^{\text{rd}}$ (or $4^{\text{th}}$ , $5^{\text{th}}$ ).
|
||||||
|
|
||||||
|
There are many possible methods of combining fan votes and judge scores. In the first two seasons of the U.S. show, the combination was based on ranks. Season 2 concerns (due to celebrity contestant Jerry Rice who was a finalist despite very low judge scores) led to a modification to use percentages instead of ranks. Examples of these two approaches are provided in the Appendix.
|
||||||
|
|
||||||
|
In season 27, another "controversy" occurred when celebrity contestant Bobby Bones won despite consistently low judges scores. In response, starting in season 28 a slight modification to the elimination process was made. The bottom two contestants were identified using the combined judge scores and fan votes, and then during the live show the judges voted to select which of these two to eliminate. Around this same season, the producers also returned to using the method of ranks to combine judges scores with fan votes as in seasons one and two. The exact season this change occurred is not known, but it is reasonable to assume it was season 28.
|
||||||
|
|
||||||
|
Judge scores are meant to reflect which dancers are technically better, although there is some subjectivity in what makes a dance better. Fan votes are likely much more subjective, influenced by the quality of the dance, but also the popularity and charisma of the celebrity. Show producers might actually prefer, to some extent, conflicts in opinions and votes as such occurrences boost fan interest and excitement.
|
||||||
|
|
||||||
|
Data with judges scores and contestant information is provided and described below. You may choose to include additional information or other data at your discretion, but you must completely document the sources. Use the data to:
|
||||||
|
|
||||||
|
- Develop a mathematical model (or models) to produce estimated fan votes (which are unknown and a closely guarded secret) for each contestant for the weeks they competed.
|
||||||
|
|
||||||
|
- Does your model correctly estimate fan votes that lead to results consistent with who was eliminated each week? Provide measures of the consistency.
|
||||||
|
- How much certainty is there in the fan vote totals you produced, and is that certainty always the same for each contestant/week? Provide measures of your certainty for the estimates.
|
||||||
|
|
||||||
|
- Use your fan vote estimates with the rest of the data to:
|
||||||
|
|
||||||
|
○ Compare and contrast the results produced by the two approaches used by the show to combine judge and fan votes (i.e. rank and percentage) across seasons (i.e. apply both approaches to each season). If differences in outcomes exist, does one method seem to favor fan votes more than the other?
|
||||||
|
○ Examine the two voting methods applied to specific celebrities where there was “controversy”, meaning differences between judges and fans. Would the choice of method to combine judge scores and fan votes have led to the same result for each of these contestants? How would including the additional approach of having judges choose which of the bottom two couples to eliminate each week impact the results? Some examples you might consider (there may also be others you identified):
|
||||||
|
|
||||||
|
- season 2 - Jerry Rice, runner up despite the lowest judges scores in 5 weeks.
|
||||||
|
- season 4 - Billy Ray Cyrus was $5^{\text{th}}$ despite last place judge scores in 6 weeks.
|
||||||
|
- season 11 - Bristol Palin was $3^{\text{rd}}$ with the lowest judge scores 12 times.
|
||||||
|
- season 27 - Bobby Bones won the despite consistently low judges scores
|
||||||
|
|
||||||
|
- Based on your analysis, which of the two methods would you recommend using for future seasons and why? Would you suggest including the additional approach of judges choosing from the bottom two couples?
|
||||||
|
|
||||||
|
- Use the data including your fan vote estimates to develop a model that analyzes the impact of various pro dancers as well as characteristics for the celebrities available in the data (age, industry, etc). How much do such things impact how well a celebrity will do in the competition? Do they impact judges scores and fan votes in the same way?
|
||||||
|
- Propose another system using fan votes and judge scores each week that you believe is more "fair" (or "better" in some other way such as making the show more exciting for the fans). Provide support for why your approach should be adopted by the show producers.
|
||||||
|
- Produce a report of no more than 25 pages with your findings and include a one- to two-page memo summarizing your results with advice for producers of DWTS on the impact of how judge and fan votes are combined with recommendations for how to do so in future seasons.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One- to two-page memo.
|
||||||
|
- References list.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
Data File: 2026_MCM_Problem_C_Data.csv – contestant information, results, and judges scores by week for seasons 1 – 34. The data description is provided in Table 1.
|
||||||
|
|
||||||
|
Table 1: Data Description for 2026_MCM_Problem_C_Data.csv
|
||||||
|
|
||||||
|
<table><tr><td>Variables</td><td>Explanation</td><td>Example</td></tr><tr><td>celebrity_name</td><td>Name of celebrity contestant (Star)</td><td>Jerry Rice, Mark Cuban, ...</td></tr><tr><td>ballroompartner</td><td>Name of professional dancer partner</td><td>Cheryl Burke, Derek Hough, ...</td></tr><tr><td>celebrity_industry</td><td>Star profession category</td><td>Athlete, Model, ...</td></tr><tr><td>celebrity_homestate</td><td>Star home state (if from U.S.)</td><td>Ohio, Maine, ...</td></tr><tr><td>celebrity_homecountry/region</td><td>Star home country/region</td><td>United States, England, ...</td></tr><tr><td>celebrity_age during season</td><td>Age of the star in the season</td><td>32, 29, ...</td></tr><tr><td>season</td><td>Season of the show</td><td>1, 2, 3, ..., 32</td></tr><tr><td>results</td><td>Season results for the start</td><td>1st Place, Eliminated Week 2, ...</td></tr><tr><td>placement</td><td>Final place for the season (1 best)</td><td>1, 2, 3, ...</td></tr><tr><td>weekXjudgeY_score</td><td>Score from judge Y in week X</td><td>1, 2, 3, ...</td></tr></table>
|
||||||
|
|
||||||
|
# Notes on the data:
|
||||||
|
|
||||||
|
1. Judges scores for each dance are from 1 (low) to 10 (high).
|
||||||
|
|
||||||
|
a. In some weeks the score reported includes a decimal (e.g. 8.5) because each celebrity performed more than one dance and the scores from each are averaged.
|
||||||
|
b. In some weeks, bonus points were awarded (dance offs etc); they are spread evenly across judge/dance scores.
|
||||||
|
c. Team dance scores were averaged with scores for each individual team member.
|
||||||
|
|
||||||
|
2. Judges are listed in the order they scored dances; thus "Judge Y" may not be the same judge from week to week, or season to season.
|
||||||
|
|
||||||
|
3. The number of celebrities is not the same across the seasons, nor is the number of weeks the show ran.
|
||||||
|
4. Season 15 was the only season to feature an all-star cast of returning celebrities.
|
||||||
|
5. There are occasionally weeks when no celebrity was eliminated, and others where more than one was eliminated.
|
||||||
|
6. $N / A$ values occur in the data set for
|
||||||
|
a. the $4^{th}$ judge score if there is not $4^{th}$ judge for that week (usually there are 3) and
|
||||||
|
b. in weeks that the show did not run in a season (for example, season 1 lasted 6 weeks so $N / A$ values are recorded for weeks 7 thru 11).
|
||||||
|
7. A 0 score is recorded for celebrities who are eliminated. For example, in Season 1 the first celebrity eliminated was Trista Sutter at the end of the Week 2 show. She thus has scores of 0 for the rest of the season (week 3 through week 6).
|
||||||
|
|
||||||
|
# Appendix: Examples of Voting Schemes
|
||||||
|
|
||||||
|
# 1. COMBINED BY RANK (used in seasons 1, 2, and $28^{\mathrm{a}}$ - 34)
|
||||||
|
|
||||||
|
In seasons 1 and 2 judges and fan votes were combined by rank. For example, in season 1, week 4 there were four remaining contestants. Rachel Hunter was eliminated meaning she received the lowest combined rank. In Table 2 the judges scores and ranks are shown, and we created one possible set of fan votes that would produce the correct result. There are many possible values for fan votes that would also give the same results. You should not use these as actual values as this is just one example. Since Rachel was ranked $2^{\text{nd}}$ by judges, in order to finish with the lowest combined score, she has the lowest fan vote ( $4^{\text{th}}$ place) for a total rank of 6.
|
||||||
|
|
||||||
|
Table 2: Example of Combining Judge and Fan Votes by Rank (Season 1, Week 4)
|
||||||
|
|
||||||
|
<table><tr><td>Contestant</td><td>Total Judges Score</td><td>Judges Score Rank</td><td>Fan Vote*</td><td>Fan Rank*</td><td>Sum of ranks</td></tr><tr><td>Rachel Hunter</td><td>25</td><td>2</td><td>1.1 million</td><td>4</td><td>6</td></tr><tr><td>Joey McIntyre</td><td>20</td><td>4</td><td>3.7 million</td><td>1</td><td>5</td></tr><tr><td>John O’Hurley</td><td>21</td><td>3</td><td>3.2 million</td><td>2</td><td>5</td></tr><tr><td>Kelly Monaco</td><td>26</td><td>1</td><td>2 million</td><td>3</td><td>4</td></tr></table>
|
||||||
|
|
||||||
|
* Fan vote/rank are unknown, hypothetical values chosen to produce the correct final ranks
|
||||||
|
|
||||||
|
# 2. COMBINED BY PERCENT (used for season 3 through $27^{\mathrm{a}}$ )
|
||||||
|
|
||||||
|
Starting in season 3 scores were combined using percents instead of ranks. An example is shown using week 9 of season 5. In that week, Jennie Garth was eliminated. Again, we artificially created fan votes that produce total percents to correctly lead to that result. The judges' percent is computed by dividing the total judge score for the contestant by the sum of total judge scores for all 4 contestants. Based on the judges' percent, Jennie was $3^{\text{rd}}$ . However, adding the percent of the 10 million artificially created fan votes we assigned to the judges' percent she was $4^{\text{th}}$ .
|
||||||
|
|
||||||
|
Table 3: Example of Combining Judge and Fan Votes by Percent (Season 5, Week 9)
|
||||||
|
|
||||||
|
<table><tr><td>Contestant</td><td>Total Judges Score</td><td>Judges Score Percent</td><td>Fan Vote*</td><td>Fan Percent*</td><td>Sum of Percent</td></tr><tr><td>Jennie Garth</td><td>29</td><td>29/117 = 24.8%</td><td>1.1 million</td><td>1.1/10 = 11%</td><td>35.8</td></tr><tr><td>Marie Osmond</td><td>28</td><td>28/117 = 23.9%</td><td>3.7 million</td><td>3.7/10 = 37%</td><td>60.9</td></tr><tr><td>Mel B</td><td>30</td><td>30/117 = 25.6%</td><td>3.2 million</td><td>3.2/10 = 32%</td><td>57.8</td></tr><tr><td>Helio Castroneves</td><td>30</td><td>30/117 = 25.6%</td><td>2 million</td><td>2/10 = 20%</td><td>45.6</td></tr><tr><td>Total</td><td>117</td><td></td><td>10 million</td><td></td><td></td></tr></table>
|
||||||
|
|
||||||
|
* Fan vote is unknown, values hypothetical to produce the correct final standings
|
||||||
|
|
||||||
|
a The year of the return to the rank based method is not known for certain; season 28 is a reasonable assumption.
|
||||||
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|||||||
|
# 2026 MCM
|
||||||
|
|
||||||
|
2026 年 MCM（Mathematical Contest in Modeling，数学建模竞赛）
|
||||||
|
|
||||||
|
# Problem A: Modeling Smartphone Battery Drain
|
||||||
|
|
||||||
|
题目 A：智能手机电池耗电建模
|
||||||
|
|
||||||
|
Smartphones are indispensable tools in modern life, yet their battery behavior often seems unpredictable. On some days a phone may last the whole day; on other days it drains rapidly before lunch. Although some users attribute this to "heavy use," the true drivers of battery depletion are more complex. Power consumption depends on the interplay of screen size and brightness, processor load, network activity, and background applications that continue drawing energy even when the device appears idle. Environmental conditions such as temperature further complicate matters: some batteries lose effective capacity in cold weather and may overheat under sustained heavy use. A battery's behavior is also influenced by its history and how it has been charged during its lifetime.
|
||||||
|
|
||||||
|
智能手机已成为现代生活中不可或缺的工具，但其电池表现却常常显得难以预测。有时手机能续航一整天；有时却在午饭前就迅速掉电。尽管一些用户将其归因于“重度使用”，电量消耗的真实驱动因素要复杂得多。功耗取决于屏幕尺寸与亮度、处理器负载、网络活动以及后台应用等多种因素的相互作用——即便设备看似空闲，后台进程仍可能持续耗电。温度等环境条件会进一步增加复杂性：部分电池在寒冷环境下有效容量下降，而在持续高负载使用时可能出现过热。电池的表现还受到其使用历史以及整个生命周期充电方式的影响。
|
||||||
|
|
||||||
|
Your task is to develop a continuous-time mathematical model of a smartphone's battery that returns the state of charge (SOC) as a function of time under realistic usage conditions. This will be used to predict the remaining time-to-empty under different conditions. You should assume that the phone has a lithium-ion battery.
|
||||||
|
|
||||||
|
你的任务是建立一个智能手机电池的连续时间数学模型，在真实使用条件下给出电池荷电状态（SOC，state of charge）随时间变化的函数关系。该模型将用于在不同条件下预测电池剩余可用时长（直至耗尽的时间）。请假设手机使用锂离子电池。
|
||||||
|
|
||||||
|
# Requirements:
|
||||||
|
|
||||||
|
要求：
|
||||||
|
|
||||||
|
1. Continuous-Time Model: Develop a model to represent the state of charge using a continuous-time equation or system of equations. You may want to begin with the simplest reasonable description of battery drain and then extend it to incorporate additional contributors such as screen usage, processor load, network connections, GPS usage, and other background tasks.
|
||||||
|
|
||||||
|
连续时间模型：使用连续时间方程或方程组建立模型来描述荷电状态（SOC）。你可以从对电池耗电的最简合理描述出发，然后扩展以纳入屏幕使用、处理器负载、网络连接、GPS 使用以及其他后台任务等额外贡献因素。
|
||||||
|
|
||||||
|
Data as support, not substitute: You may collect or use data for parameter estimation and validation. If open datasets are limited, you may use published measurements or specifications (with proper citation), provided parameters are clearly justified and validated for plausibility. However, projects based solely on discrete curve fitting, timestep regression, or black-box machine learning without an explicit continuous-time model will not satisfy this problem's requirements. All data used must be well documented and freely available, and the data must be free for use under an open license.
|
||||||
|
|
||||||
|
数据用于支撑，而非替代建模：你可以收集或使用数据进行参数估计与验证。若公开数据集有限，可使用已发表的测量结果或技术规格（须规范引用），但需对参数进行清晰论证，并验证其合理性。然而，仅基于离散曲线拟合、时间步回归，或缺乏明确连续时间模型的黑箱机器学习方法的项目，不符合本题要求。所有使用的数据必须有充分文档说明并可自由获取，且须在开放许可下允许自由使用。
|
||||||
|
|
||||||
|
2. Time-to-Empty predictions: Use your model to compute or approximate the time-to-empty under various initial charge levels and usage scenarios. Compare predictions to observed or plausible behavior, quantify uncertainty, and identify where the model performs well or poorly.
|
||||||
|
|
||||||
|
耗尽时间预测：使用你的模型在不同初始电量与不同使用情景下计算或近似得到“电量耗尽所需时间”。将预测结果与观测到的行为或合理的典型表现进行比较，量化不确定性，并指出模型在哪些方面表现良好、哪些方面表现欠佳。
|
||||||
|
|
||||||
|
- Show how your model explains differences in these outcomes and identify the specific drivers of rapid battery drain in each case.
|
||||||
|
|
||||||
|
展示你的模型如何解释上述不同结果之间的差异，并识别在每种情形下导致电量快速下降的具体驱动因素。
|
||||||
|
|
||||||
|
- Which activities or conditions produce the greatest reductions in battery life? Which ones change the model surprisingly little?
|
||||||
|
|
||||||
|
哪些活动或条件会使电池续航时间下降最多？哪些因素对模型输出的影响出乎意料地小？
|
||||||
|
|
||||||
|
3. Sensitivity and Assumptions: Examine how your predictions vary after making changes in your modeling assumptions, parameter values, and fluctuations in usage patterns.
|
||||||
|
|
||||||
|
敏感性与假设：研究当你的建模假设、参数取值以及使用模式的波动发生变化时，预测结果将如何随之改变。
|
||||||
|
|
||||||
|
4. Recommendations: Translate your findings into practical recommendations for a cellphone user. For example, which user behaviors—such as reducing brightness, disabling background tasks, or switching network modes—yield the largest improvements in battery life? How might an operating system implement more effective power-saving strategies based on insights from your model? Consider how battery aging reduces effective capacity or how your modeling framework could generalize to other portable devices.
|
||||||
|
|
||||||
|
建议：将你的发现转化为对手机用户具有可操作性的建议。例如，哪些用户行为——如降低亮度、禁用后台任务或切换网络模式——能带来最大的续航提升？操作系统可如何基于你模型的洞见实施更有效的省电策略？请考虑电池老化如何降低有效容量，以及你的建模框架如何推广到其他便携式设备。
|
||||||
|
|
||||||
|
# Your report should present:
|
||||||
|
|
||||||
|
报告应包含：
|
||||||
|
|
||||||
|
- A clear description of your model and governing equations.
|
||||||
|
|
||||||
|
对你的模型及其控制方程的清晰描述。
|
||||||
|
|
||||||
|
- The assumptions and rationale behind your design choices.
|
||||||
|
|
||||||
|
支撑你设计选择的假设与理由。
|
||||||
|
|
||||||
|
- Parameter estimation methods and validation results.
|
||||||
|
|
||||||
|
参数估计方法与验证结果。
|
||||||
|
|
||||||
|
- A discussion of strengths, limitations, and possible extensions.
|
||||||
|
|
||||||
|
对模型优势、局限性与可扩展方向的讨论。
|
||||||
|
|
||||||
|
- An executive-style summary highlighting main results, insights, and recommendations.
|
||||||
|
|
||||||
|
一份“高层摘要”式的总结，突出主要结果、洞见与建议。
|
||||||
|
|
||||||
|
Important: Your model must be grounded in clearly defined physical or mechanical reasoning; discrete curve fitting or other mathematical forms that are disconnected from an explicit continuous-time description of battery behavior will not satisfy the requirements. Projects that rely solely on discrete curve fitting or statistical regression without a clearly formulated continuous-time model will not satisfy the requirements of this problem.
|
||||||
|
|
||||||
|
重要提示：你的模型必须建立在清晰界定的物理或机理性推理基础之上；若仅进行离散曲线拟合，或采用与电池行为的明确连续时间描述脱节的其他数学形式，将不满足要求。仅依赖离散曲线拟合或统计回归、且未清晰提出连续时间模型的项目，同样不符合本题要求。
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
你的 PDF 解决方案（总页数不超过 25 页）应包含：
|
||||||
|
|
||||||
|
- One-page Summary Sheet.
|
||||||
|
|
||||||
|
1 页摘要页（Summary Sheet）。
|
||||||
|
|
||||||
|
- Table of Contents.
|
||||||
|
|
||||||
|
目录。
|
||||||
|
|
||||||
|
- Your complete solution.
|
||||||
|
|
||||||
|
完整解答。
|
||||||
|
|
||||||
|
- In-text Citations and A Reference List.
|
||||||
|
|
||||||
|
文内引用与参考文献列表。
|
||||||
|
|
||||||
|
- AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
AI 使用报告（如使用；不计入 25 页总页数限制）。
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
注：完整的 MCM 提交稿没有规定最低页数要求。你的全部解题内容及任何补充信息（例如：图示、示意图、计算、表格）最多可使用 25 页。允许提交部分解答。竞赛允许谨慎使用 ChatGPT 等 AI 工具，但完成本题并不依赖 AI。如你选择使用生成式 AI，必须遵循 COMAP 的 AI 使用政策；这将要求你在 PDF 解答文件末尾附加一份 AI 使用报告，该报告不计入 25 页总页数限制。
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
术语表
|
||||||
|
|
||||||
|
Smartphone: is a mobile device that combines the functionality of a traditional cell phone with advanced computing capabilities.
|
||||||
|
|
||||||
|
智能手机：一种将传统手机通信功能与高级计算能力相结合的移动设备。
|
||||||
|
|
||||||
|
Power Consumption: the rate at which a device uses electrical energy from its battery or power source.
|
||||||
|
|
||||||
|
功耗：设备从电池或电源中消耗电能的速率。
|
||||||
|
|
||||||
|
Processor Load: the actual amount of work being done by the processor at a given moment.
|
||||||
|
|
||||||
|
处理器负载：处理器在某一时刻实际执行的工作量水平。
|
||||||
|
|
||||||
|
State of Charge (SOC): a measure of how much energy remains in a battery compared to its full capacity, expressed as a percentage.
|
||||||
|
|
||||||
|
荷电状态（SOC）：电池剩余能量相对于其满容量的比例度量，通常以百分比表示。
|
||||||
|
|
||||||
|
Time-to-Empty: the estimated amount of time remaining before a battery is completely discharged.
|
||||||
|
|
||||||
|
耗尽时间（Time-to-Empty）：电池完全放电之前预计剩余的时间长度。
|
||||||
119
B_zh_en.md
Normal file
119
B_zh_en.md
Normal file
@@ -0,0 +1,119 @@
|
|||||||
|
# 2026 MCM
|
||||||
|
|
||||||
|
2026 年 MCM（Mathematical Contest in Modeling，数学建模竞赛）
|
||||||
|
|
||||||
|
# Problem B: Creating a Moon Colony Using a Space Elevator System
|
||||||
|
|
||||||
|
题目 B：利用空间电梯系统建设月球殖民地
|
||||||
|
|
||||||
|
Imagine a future where it's possible for anyone to visit space by taking a leisurely and scenic ride from the Equator to Earth's orbit and then catching a routine, safe, and inexpensive rocket flight to the Moon, Mars, or beyond. In this future, we could build lush, green, and beautiful space habitats with artificial gravity, where people would vacation, work, or even live. These habitats would alleviate pressure on Earth's delicate, overworked, and fragile ecosystems. The technology to enable these events would provide humankind with limitless, safe, routine, environmentally friendly, efficient, and global access to space. To achieve these goals, some people envision a Space Elevator System, powered by electricity, offering a scalable infrastructure for interplanetary logistics, commerce, and exploration.
|
||||||
|
|
||||||
|
设想这样一个未来：任何人都可以从赤道出发，乘坐一段轻松且风景优美的旅程抵达地球轨道，随后再搭乘一趟常态化、安全且廉价的火箭飞行前往月球、火星乃至更远的深空。在这样的未来里，我们能够建造拥有人工重力、郁郁葱葱且美丽宜居的太空栖居地，人们可以在那里度假、工作，甚至长期居住。这些栖居地将缓解地球脆弱且超负荷运转的生态系统所承受的压力。实现上述愿景所需的技术，将为人类提供几乎无限、安全、常态化、环境友好、高效且面向全球的太空通达能力。为达成这些目标，有人提出构建一套以电力驱动的“空间电梯系统”，为行星际物流、商业与探索提供可扩展的基础设施。
|
||||||
|
|
||||||
|
At its final operating configuration, the Space Elevator System would comprise three Galactic Harbours, ideally separated by 120 degrees around the equator. Each Galactic Harbour would include a single Earth port with two $100,000\mathrm{km}$ -long tethers connected to two apex anchors, with multiple space elevators operating together, each capable of lifting massive payloads daily from Earth to geosynchronous orbit (GEO) and beyond to the apex anchor where they can be loaded on a rocket and delivered anywhere using much less fuel.
|
||||||
|
|
||||||
|
在最终运行配置下，空间电梯系统将由三个 Galactic Harbour（可译作“银河港”）构成，理想情况下沿赤道周向相隔 120 度。每个银河港包含一个地面港口（Earth Port），并通过两条长度为 $100,000\mathrm{km}$ 的缆索（tether）分别连接到两个顶点锚（apex anchor）。多个空间电梯将协同运行，每部电梯都能够每天将巨型载荷从地面提升至地球同步轨道（GEO）并继续上行至顶点锚；载荷可在顶点锚处装载到火箭上，并以远低于传统方式所需的燃料消耗运往任意目的地。
|
||||||
|
|
||||||
|
The Moon Colony Management (MCM) Agency is preparing to build a Moon Colony with an estimated 100,000 people beginning in the year 2050, after completion of the Space Elevator System. It is estimated that the Moon Colony will need about 100 million metric tons of materials. Additionally, water and supplies will routinely need to be sent to sustain the Moon's population once the colony is complete. To get to the Moon, the Galactic Harbour must send material in two steps: first, from the Earth port to the apex anchor via a space elevator, and second, from the apex anchor to the Moon Colony via a rocket. The MCM Agency anticipates that the Galactic Harbour will provide an advanced lift system capable of moving 179,000 metric tons every year, while generating no atmospheric pollution.
|
||||||
|
|
||||||
|
月球殖民地管理机构（Moon Colony Management，MCM）计划在空间电梯系统建成后，自 2050 年起建设一座预计可容纳 100,000 人的月球殖民地。据估计，月球殖民地建设将需要约 1 亿公吨材料。此外，在殖民地建成并投入运行后，还需要持续、常态化地运送水与各类补给以维持月球人口。将物资运抵月球需经由银河港分两步完成：第一步，通过空间电梯将物资从地面港口运至顶点锚；第二步，再由火箭将物资从顶点锚运至月球殖民地。MCM 机构预计，银河港将提供一种先进的提升系统，年运输能力可达 179,000 公吨，同时不产生大气污染。
|
||||||
|
|
||||||
|
The agency is also considering using traditional rockets to supply material for construction and supplies to the Moon Colony. The Earth current has ten rocket launch sites: Alaska, California, Texas, Florida, and Virginia (United States), Kazakhstan, French Guiana, Satish Dhawan Space Centre (India), Taiyuan Satellite Launch Center (China), and Mahia Peninsula (New Zealand).
|
||||||
|
|
||||||
|
该机构也在考虑采用传统火箭向月球殖民地运送建设材料与补给。地球目前有十个火箭发射场：阿拉斯加、加利福尼亚、得克萨斯、佛罗里达、弗吉尼亚（美国），哈萨克斯坦，法属圭亚那，萨迪什·达万航天中心（印度），太原卫星发射中心（中国），以及马希亚半岛（新西兰）。
|
||||||
|
|
||||||
|
A rocket would require a single step from the rocket launch site on Earth to the Moon Colony. By 2050 it is estimated that rockets will be able to carry 100-150 metric tons of payload to the Moon using advanced Falcon Heavy launches. You may assume perfect conditions for both the Galactic Harbour system (e.g., no swaying of the tether) and rocket launches (e.g., no failed launches). You should consider the cost and timeline to deliver the materials from the surface of the Earth to the Moon Colony site for the different scenarios.
|
||||||
|
|
||||||
|
使用火箭时，只需一步即可将载荷从地球发射场直接运至月球殖民地。预计到 2050 年，借助先进的 Falcon Heavy 发射任务，火箭将能够向月球运送 100–150 公吨的有效载荷。你可以假设银河港系统与火箭发射均处于理想条件（例如缆索不发生摆动、发射无失败）。你需要在不同情景下，综合考虑将材料从地球表面运至月球殖民地选址的成本与时间进度。
|
||||||
|
|
||||||
|
# Your Task:
|
||||||
|
|
||||||
|
你的任务：
|
||||||
|
|
||||||
|
Your task is to utilize a mathematical model to determine the cost and associated timeline in order to transport material to build a 100,000 person Moon Colony starting in 2050. You will need to compare the Modern-Day Space Elevator System's three Galactic Harbours to traditional rockets launched from selected rocket bases.
|
||||||
|
|
||||||
|
你的任务是建立并运用数学模型，确定自 2050 年起为建设一座可容纳 100,000 人的月球殖民地所需的物资运输成本及相应时间进度。你需要比较“现代空间电梯系统”的三座银河港与从选定火箭发射基地发射的传统火箭方案。
|
||||||
|
|
||||||
|
# Your model should include:
|
||||||
|
|
||||||
|
你的模型应包含：
|
||||||
|
|
||||||
|
1. Consideration of three different scenarios for how the 100 million metric tons of materials will be delivered to build the 100,000-person Moon Colony;
|
||||||
|
|
||||||
|
考虑用于向月球殖民地交付 1 亿公吨材料（以建设可容纳 100,000 人的月球殖民地）的三种不同情景；
|
||||||
|
|
||||||
|
a. using the Space Elevator System's three Galactic Harbor's alone,
|
||||||
|
仅使用空间电梯系统的三座银河港；
|
||||||
|
b. traditional rocket launches from existing bases alone (you may choose which facilities to use), or,
|
||||||
|
仅使用现有发射场的传统火箭发射（你可自行选择使用哪些设施）；或
|
||||||
|
c. some combination of the two methods.
|
||||||
|
两种方式的某种组合。
|
||||||
|
|
||||||
|
2. To what extent does your solution(s) change if the transportation systems are not in perfect working order (e.g, swaying of the tether, rockets fail, elevators break, etc.).?
|
||||||
|
|
||||||
|
如果运输系统并非完美运行（例如缆索摆动、火箭发射失败、电梯故障等），你的解（或多种解）在多大程度上会发生变化？
|
||||||
|
|
||||||
|
3. Investigate the water needs for a one-year period once the 100,000-person Moon Colony is fully operational. Use your delivery model to understand the additional cost and timeline needed to ensure the colony has sufficient water for one full year after the Moon Colony is inhabited.
|
||||||
|
|
||||||
|
当 100,000 人规模的月球殖民地全面投入运行后，研究其一年期的用水需求。运用你的交付模型分析：为确保殖民地在开始有人居住后的一整年内有充足用水所需的额外成本与时间进度。
|
||||||
|
|
||||||
|
4. Discuss the impact on the Earth's environment for achieving the 100,000-person Moon Colony under the different scenarios. How would you adjust your model to minimize the environmental impact?
|
||||||
|
|
||||||
|
讨论在不同情景下实现 100,000 人月球殖民地对地球环境的影响。你将如何调整模型以尽量降低环境影响？
|
||||||
|
|
||||||
|
5. Write a one-page letter recommending a course of action to the fictional MCM Agency to build and sustain a 100,000-person Moon Colony.
|
||||||
|
|
||||||
|
撰写一封一页的建议信，向虚构的 MCM 机构推荐建设并维持一座可容纳 100,000 人的月球殖民地的行动方案。
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
你的 PDF 解答（总页数不超过 25 页）应包括：
|
||||||
|
|
||||||
|
- One-page Summary Sheet.
|
||||||
|
1 页摘要页（Summary Sheet）。
|
||||||
|
- Table of Contents.
|
||||||
|
目录。
|
||||||
|
- Your complete solution.
|
||||||
|
完整解答。
|
||||||
|
- One-page letter to MCM Agency
|
||||||
|
致 MCM 机构的一页建议信。
|
||||||
|
- References list.
|
||||||
|
参考文献列表。
|
||||||
|
- AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
AI 使用报告（如使用；不计入 25 页总页数限制。）
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
注：完整的 MCM 提交稿没有规定最低页数要求。你的全部解题内容及任何补充信息（例如：图示、示意图、计算、表格）最多可使用 25 页。允许提交部分解答。竞赛允许谨慎使用 ChatGPT 等 AI 工具，但完成本题并不依赖 AI。如你选择使用生成式 AI，必须遵循 COMAP 的 AI 使用政策；这将要求你在 PDF 解答文件末尾附加一份 AI 使用报告，该报告不计入 25 页总页数限制。
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
术语表
|
||||||
|
|
||||||
|
Space Elevator System is comprised of three Galactic Harbours plus additional support facilities.
|
||||||
|
|
||||||
|
空间电梯系统：由三座银河港及其他配套支撑设施组成。
|
||||||
|
|
||||||
|
Galactic Harbour is comprised of two apex anchors each connected by two tethers to a single Earth Port.
|
||||||
|
|
||||||
|
银河港（Galactic Harbour）：由两个顶点锚组成；每个顶点锚通过两条缆索连接到同一个地面港口（Earth Port）。
|
||||||
|
|
||||||
|
Earth Port is the location on the Earth that provides surface support for the Galactic Harbour.
|
||||||
|
|
||||||
|
地面港口（Earth Port）：在地球表面为银河港提供地面支撑与保障的地点。
|
||||||
|
|
||||||
|
Tethers are $100,000\mathrm{km}$ long graphene material that links the Earth port and apex anchors in the Space Elevator System.
|
||||||
|
|
||||||
|
缆索（tether）：空间电梯系统中连接地面港口与顶点锚的石墨烯材料缆索，长度为 $100,000\mathrm{km}$。
|
||||||
|
|
||||||
|
Apex Anchor is the counterweight in space at the end of the $100,000\mathrm{km}$ tether.
|
||||||
|
|
||||||
|
顶点锚（apex anchor）：位于 $100,000\mathrm{km}$ 缆索末端、在太空中起配重作用的结构。
|
||||||
|
|
||||||
|
Geosynchronous orbit (GEO) is approximately $35,786\mathrm{km}$ above the surface of the Earth where the orbital period to circle Earth is 24 hours, matching Earth's rotation so it stays over the same longitude each day.
|
||||||
|
|
||||||
|
地球同步轨道（GEO，geosynchronous orbit）：距离地球表面约 $35,786\mathrm{km}$ 的轨道；其绕地周期为 24 小时，与地球自转周期一致，因此每天可保持位于同一经度上空。
|
||||||
|
|
||||||
|
Moon Colony is a habitat on the moon with the capacity to support 100,000-persons.
|
||||||
|
|
||||||
|
月球殖民地（Moon Colony）：位于月球、可支持 100,000 人居住与生活的栖居地。
|
||||||
231
C_zh_en.md
Normal file
231
C_zh_en.md
Normal file
@@ -0,0 +1,231 @@
|
|||||||
|
# 2026 MCM Problem C: Data With The Stars
|
||||||
|
|
||||||
|
2026 MCM 题目 C：与星共舞的数据
|
||||||
|
|
||||||
|
Dancing with the Stars (DWTS) is the American version of an international television franchise based on the British show "Strictly Come Dancing" ("Come Dancing" originally). Versions of the show have appeared in Albania, Argentina, Australia, China, France, India, and many other countries. The U.S. version, the focus of this problem, has completed 34 seasons.
|
||||||
|
|
||||||
|
《与星共舞》（Dancing with the Stars，DWTS）是一个国际电视节目品牌的美国版，源自英国节目 “Strictly Come Dancing”（其前身为 “Come Dancing”）。该节目已在阿尔巴尼亚、阿根廷、澳大利亚、中国、法国、印度等多个国家推出不同版本。本题聚焦的美国版已播出 34 季。
|
||||||
|
|
||||||
|
Celebrities are partnered with professional dancers and then perform dances each week. A panel of expert judges scores each couple's dance, and fans vote (by phone or online) for their favorite couple that week. Fans can vote once or multiple times up to a limit announced each week. Further, fans vote for the star they wish to keep, but cannot vote to eliminate a star. The judge and fan votes are combined in order to determine which couple to eliminate (the lowest combined score) that week. Three (in some seasons more) couples reach the finals and in the week of the finals the combined scores from fans and judges are used to rank them from $1^{\text{st}}$ to $3^{\text{rd}}$ (or $4^{\text{th}}$ , $5^{\text{th}}$ ).
|
||||||
|
|
||||||
|
明星与专业舞者配对，每周进行舞蹈表演。由专家评委团为每对选手评分，同时观众通过电话或网络为当周最喜欢的组合投票。观众每周可以投一次或多次票，但有当周公布的投票上限。此外，观众只能投票支持希望保留的明星，不能投票“淘汰”某位明星。评委评分与观众投票会被合并以确定当周被淘汰的组合（合并得分最低者）。最终有三对（部分季数更多）进入决赛，在决赛周，评委与观众的合并得分用于将他们从 $1^{\text{st}}$ 排名到 $3^{\text{rd}}$ （或 $4^{\text{th}}$ 、 $5^{\text{th}}$ ）。
|
||||||
|
|
||||||
|
There are many possible methods of combining fan votes and judge scores. In the first two seasons of the U.S. show, the combination was based on ranks. Season 2 concerns (due to celebrity contestant Jerry Rice who was a finalist despite very low judge scores) led to a modification to use percentages instead of ranks. Examples of these two approaches are provided in the Appendix.
|
||||||
|
|
||||||
|
合并观众投票与评委评分有多种可能方式。在美国版前两季，合并方式以“排名”为基础。第二季因名人选手 Jerry Rice 尽管评委评分很低仍进入决赛而引发争议，促使规则改为使用“百分比”而非“排名”。附录提供了这两种方法的示例。
|
||||||
|
|
||||||
|
In season 27, another "controversy" occurred when celebrity contestant Bobby Bones won despite consistently low judges scores. In response, starting in season 28 a slight modification to the elimination process was made. The bottom two contestants were identified using the combined judge scores and fan votes, and then during the live show the judges voted to select which of these two to eliminate. Around this same season, the producers also returned to using the method of ranks to combine judges scores with fan votes as in seasons one and two. The exact season this change occurred is not known, but it is reasonable to assume it was season 28.
|
||||||
|
|
||||||
|
第 27 季出现了另一场“争议”：名人选手 Bobby Bones 尽管评委评分持续偏低却最终夺冠。为此，从第 28 季开始，淘汰流程进行了小幅调整：先依据评委评分与观众投票的合并结果确定得分最低的两对选手，然后在直播中由评委投票决定淘汰其中一对。大约在同一时期，制作方也回到了第 1、2 季使用的“排名法”来合并评委评分与观众投票。具体变更发生于哪一季并不确定，但合理的假设是第 28 季。
|
||||||
|
|
||||||
|
Judge scores are meant to reflect which dancers are technically better, although there is some subjectivity in what makes a dance better. Fan votes are likely much more subjective, influenced by the quality of the dance, but also the popularity and charisma of the celebrity. Show producers might actually prefer, to some extent, conflicts in opinions and votes as such occurrences boost fan interest and excitement.
|
||||||
|
|
||||||
|
评委评分旨在反映舞者的技术水平高低，但“何为更好”的判断仍带有一定主观性。观众投票可能更具主观性，既受舞蹈质量影响，也受明星人气与个人魅力左右。节目制作方在一定程度上可能并不排斥评委与观众之间的分歧，因为此类冲突会提升粉丝关注度与节目热度。
|
||||||
|
|
||||||
|
Data with judges scores and contestant information is provided and described below. You may choose to include additional information or other data at your discretion, but you must completely document the sources. Use the data to:
|
||||||
|
|
||||||
|
题目提供并在下文说明了评委评分与选手信息数据。你可自行选择加入其他信息或数据，但必须完整注明来源。使用这些数据来：
|
||||||
|
|
||||||
|
- Develop a mathematical model (or models) to produce estimated fan votes (which are unknown and a closely guarded secret) for each contestant for the weeks they competed.
|
||||||
|
|
||||||
|
建立数学模型（或多个模型），为每位参赛者在其参赛的各周估计观众投票数（真实票数未知且高度保密）。
|
||||||
|
|
||||||
|
- Does your model correctly estimate fan votes that lead to results consistent with who was eliminated each week? Provide measures of the consistency.
|
||||||
|
- How much certainty is there in the fan vote totals you produced, and is that certainty always the same for each contestant/week? Provide measures of your certainty for the estimates.
|
||||||
|
|
||||||
|
你的模型是否能估计出与每周淘汰结果一致的观众投票数？请提供一致性度量。
|
||||||
|
你对所生成的观众投票总数有多大置信度？这种置信度在不同选手/不同周是否一致？请给出置信度度量。
|
||||||
|
|
||||||
|
- Use your fan vote estimates with the rest of the data to:
|
||||||
|
|
||||||
|
使用你估计的观众投票并结合其余数据来：
|
||||||
|
|
||||||
|
○ Compare and contrast the results produced by the two approaches used by the show to combine judge and fan votes (i.e. rank and percentage) across seasons (i.e. apply both approaches to each season). If differences in outcomes exist, does one method seem to favor fan votes more than the other?
|
||||||
|
○ Examine the two voting methods applied to specific celebrities where there was “controversy”, meaning differences between judges and fans. Would the choice of method to combine judge scores and fan votes have led to the same result for each of these contestants? How would including the additional approach of having judges choose which of the bottom two couples to eliminate each week impact the results? Some examples you might consider (there may also be others you identified):
|
||||||
|
|
||||||
|
○ 比较并对照节目使用的两种合并评委与观众投票的方法（即排名法与百分比法）在各季产生的结果（即对每一季都应用两种方法）。若结果存在差异，是否有一种方法相对更偏向观众投票？
|
||||||
|
○ 针对存在“争议”的特定名人（即评委与观众意见存在明显分歧者）比较两种投票方法。不同的合并方法会否导致这些选手的结果相同？若再加入“评委从最后两名中选择淘汰对象”的额外机制，又会如何影响结果？你可参考以下案例（也可加入你识别的其他案例）：
|
||||||
|
|
||||||
|
- season 2 - Jerry Rice, runner up despite the lowest judges scores in 5 weeks.
|
||||||
|
- season 4 - Billy Ray Cyrus was $5^{\text{th}}$ despite last place judge scores in 6 weeks.
|
||||||
|
- season 11 - Bristol Palin was $3^{\text{rd}}$ with the lowest judge scores 12 times.
|
||||||
|
- season 27 - Bobby Bones won the despite consistently low judges scores
|
||||||
|
|
||||||
|
第 2 季——Jerry Rice：尽管有 5 周评委最低分，仍获得亚军。
|
||||||
|
第 4 季——Billy Ray Cyrus：尽管有 6 周评委排名末位，仍获得 $5^{\text{th}}$ 。
|
||||||
|
第 11 季——Bristol Palin：评委最低分出现 12 次，最终排名 $3^{\text{rd}}$ 。
|
||||||
|
第 27 季——Bobby Bones：尽管评委评分持续偏低仍夺冠。
|
||||||
|
|
||||||
|
- Based on your analysis, which of the two methods would you recommend using for future seasons and why? Would you suggest including the additional approach of judges choosing from the bottom two couples?
|
||||||
|
|
||||||
|
基于你的分析，你会推荐未来赛季使用哪一种方法，为什么？你是否建议加入“评委从最后两名中选择淘汰对象”的额外机制？
|
||||||
|
|
||||||
|
- Use the data including your fan vote estimates to develop a model that analyzes the impact of various pro dancers as well as characteristics for the celebrities available in the data (age, industry, etc). How much do such things impact how well a celebrity will do in the competition? Do they impact judges scores and fan votes in the same way?
|
||||||
|
- Propose another system using fan votes and judge scores each week that you believe is more "fair" (or "better" in some other way such as making the show more exciting for the fans). Provide support for why your approach should be adopted by the show producers.
|
||||||
|
- Produce a report of no more than 25 pages with your findings and include a one- to two-page memo summarizing your results with advice for producers of DWTS on the impact of how judge and fan votes are combined with recommendations for how to do so in future seasons.
|
||||||
|
|
||||||
|
使用包括你所估计观众投票在内的数据，建立模型分析专业舞者与名人特征（年龄、行业等）对比赛成绩的影响。这些因素对选手表现的影响程度有多大？它们对评委评分与观众投票的影响是否一致？
|
||||||
|
提出一个你认为更“公平”（或在其他方面更“好”，例如让节目更精彩）的、基于每周评委评分与观众投票的替代方案，并给出理由说明为何该方案应被制作方采纳。
|
||||||
|
形成不超过 25 页的报告，给出你的发现，并附上一至两页的备忘录，总结结果并向 DWTS 制作方提出关于评委与观众投票合并方式的建议以及未来赛季的推荐做法。
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
你的 PDF 解答（总页数不超过 25 页）应包括：
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One- to two-page memo.
|
||||||
|
- References list.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
1 页摘要页（Summary Sheet）。
|
||||||
|
目录。
|
||||||
|
- 完整解答。
|
||||||
|
一至两页的备忘录。
|
||||||
|
- 参考文献列表。
|
||||||
|
AI 使用报告（如使用；不计入 25 页总页数限制。）
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
注：完整的 MCM 提交稿没有规定最低页数要求。你的全部解题内容及任何补充信息（例如：图示、示意图、计算、表格）最多可使用 25 页。允许提交部分解答。竞赛允许谨慎使用 ChatGPT 等 AI 工具，但完成本题并不依赖 AI。如你选择使用生成式 AI，必须遵循 COMAP 的 AI 使用政策；这将要求你在 PDF 解答文件末尾附加一份 AI 使用报告，该报告不计入 25 页总页数限制。
|
||||||
|
|
||||||
|
Data File: 2026_MCM_Problem_C_Data.csv – contestant information, results, and judges scores by week for seasons 1 – 34. The data description is provided in Table 1.
|
||||||
|
|
||||||
|
数据文件：2026_MCM_Problem_C_Data.csv —— 包含第 1 至 34 季的参赛者信息、赛季结果及按周的评委评分。数据说明见表 1。
|
||||||
|
|
||||||
|
Table 1: Data Description for 2026_MCM_Problem_C_Data.csv
|
||||||
|
|
||||||
|
表 1：2026_MCM_Problem_C_Data.csv 数据说明
|
||||||
|
|
||||||
|
| Variables | Explanation | Example |
|
||||||
|
|---|---|---|
|
||||||
|
| `celebrity_name` | Name of celebrity contestant (Star) | Jerry Rice, Mark Cuban, ... |
|
||||||
|
| `ballroompartner` | Name of professional dancer partner | Cheryl Burke, Derek Hough, ... |
|
||||||
|
| `celebrity_industry` | Star profession category | Athlete, Model, ... |
|
||||||
|
| `celebrity_homestate` | Star home state (if from U.S.) | Ohio, Maine, ... |
|
||||||
|
| `celebrity_homecountry/region` | Star home country/region | United States, England, ... |
|
||||||
|
| `celebrity_age during season` | Age of the star in the season | 32, 29, ... |
|
||||||
|
| `season` | Season of the show | 1, 2, 3, ..., 32 |
|
||||||
|
| `results` | Season results for the start | 1st Place, Eliminated Week 2, ... |
|
||||||
|
| `placement` | Final place for the season (1 best) | 1, 2, 3, ... |
|
||||||
|
| `weekXjudgeY_score` | Score from judge Y in week X | 1, 2, 3, ... |
|
||||||
|
|
||||||
|
| 变量 | 说明 | 示例 |
|
||||||
|
|---|---|---|
|
||||||
|
| `celebrity_name` | 名人参赛者（明星）姓名 | Jerry Rice, Mark Cuban, ... |
|
||||||
|
| `ballroompartner` | 专业舞者搭档姓名 | Cheryl Burke, Derek Hough, ... |
|
||||||
|
| `celebrity_industry` | 明星职业类别 | Athlete, Model, ... |
|
||||||
|
| `celebrity_homestate` | 明星所在州（若来自美国） | Ohio, Maine, ... |
|
||||||
|
| `celebrity_homecountry/region` | 明星所属国家/地区 | United States, England, ... |
|
||||||
|
| `celebrity_age during season` | 该季明星年龄 | 32, 29, ... |
|
||||||
|
| `season` | 节目季数 | 1, 2, 3, ..., 32 |
|
||||||
|
| `results` | 该季结果 | 1st Place, Eliminated Week 2, ... |
|
||||||
|
| `placement` | 该季最终名次（1 为最佳） | 1, 2, 3, ... |
|
||||||
|
| `weekXjudgeY_score` | 第 X 周评委 Y 的评分 | 1, 2, 3, ... |
|
||||||
|
|
||||||
|
# Notes on the data:
|
||||||
|
|
||||||
|
数据说明：
|
||||||
|
|
||||||
|
1. Judges scores for each dance are from 1 (low) to 10 (high).
|
||||||
|
|
||||||
|
每支舞的评委评分范围为 1（低）到 10（高）。
|
||||||
|
|
||||||
|
a. In some weeks the score reported includes a decimal (e.g. 8.5) because each celebrity performed more than one dance and the scores from each are averaged.
|
||||||
|
b. In some weeks, bonus points were awarded (dance offs etc); they are spread evenly across judge/dance scores.
|
||||||
|
c. Team dance scores were averaged with scores for each individual team member.
|
||||||
|
|
||||||
|
a. 某些周的得分包含小数（例如 8.5），因为明星当周表演多支舞蹈，所报分数为多支舞的平均值。
|
||||||
|
b. 某些周会有加分（例如对决舞等）；加分会均匀分配到各评委/舞蹈评分中。
|
||||||
|
c. 团体舞得分会与团队中每位成员的个人得分进行平均。
|
||||||
|
|
||||||
|
2. Judges are listed in the order they scored dances; thus "Judge Y" may not be the same judge from week to week, or season to season.
|
||||||
|
|
||||||
|
评委按打分顺序列出，因此“Judge Y”并不一定在不同周或不同季对应同一位评委。
|
||||||
|
|
||||||
|
3. The number of celebrities is not the same across the seasons, nor is the number of weeks the show ran.
|
||||||
|
4. Season 15 was the only season to feature an all-star cast of returning celebrities.
|
||||||
|
5. There are occasionally weeks when no celebrity was eliminated, and others where more than one was eliminated.
|
||||||
|
6. $N / A$ values occur in the data set for
|
||||||
|
a. the $4^{th}$ judge score if there is not $4^{th}$ judge for that week (usually there are 3) and
|
||||||
|
b. in weeks that the show did not run in a season (for example, season 1 lasted 6 weeks so $N / A$ values are recorded for weeks 7 thru 11).
|
||||||
|
7. A 0 score is recorded for celebrities who are eliminated. For example, in Season 1 the first celebrity eliminated was Trista Sutter at the end of the Week 2 show. She thus has scores of 0 for the rest of the season (week 3 through week 6).
|
||||||
|
|
||||||
|
3. 不同季的名人数量不同，节目持续周数也不同。
|
||||||
|
4. 第 15 季是唯一一季全明星回归赛季。
|
||||||
|
5. 有时某些周没有淘汰，也有时一周淘汰多于一人。
|
||||||
|
6. 数据集中出现 $N / A$ 值的情况包括：
|
||||||
|
a. 若当周没有第 $4^{th}$ 位评委（通常只有 3 位），则第 $4^{th}$ 位评委评分为 $N / A$ ；
|
||||||
|
b. 某季中节目未播出的周（例如第 1 季仅持续 6 周，因此第 7 至第 11 周记为 $N / A$ ）。
|
||||||
|
7. 被淘汰选手之后的周次记为 0 分。例如第 1 季中首位被淘汰的是 Trista Sutter，她在第 2 周节目结束时被淘汰，因此第 3 至第 6 周的评分均为 0。
|
||||||
|
|
||||||
|
# Appendix: Examples of Voting Schemes
|
||||||
|
|
||||||
|
# 附录：投票方案示例
|
||||||
|
|
||||||
|
# 1. COMBINED BY RANK (used in seasons 1, 2, and $28^{\mathrm{a}}$ - 34)
|
||||||
|
|
||||||
|
# 1. 按排名合并（用于第 1、2 季及 $28^{\mathrm{a}}$ – 34 季）
|
||||||
|
|
||||||
|
In seasons 1 and 2 judges and fan votes were combined by rank. For example, in season 1, week 4 there were four remaining contestants. Rachel Hunter was eliminated meaning she received the lowest combined rank. In Table 2 the judges scores and ranks are shown, and we created one possible set of fan votes that would produce the correct result. There are many possible values for fan votes that would also give the same results. You should not use these as actual values as this is just one example. Since Rachel was ranked $2^{\text{nd}}$ by judges, in order to finish with the lowest combined score, she has the lowest fan vote ( $4^{\text{th}}$ place) for a total rank of 6.
|
||||||
|
|
||||||
|
在第 1、2 季，评委与观众投票按“排名”合并。以第 1 季第 4 周为例，当时剩余四名选手。Rachel Hunter 被淘汰，说明她的合并排名最低。表 2 展示了评委评分与排名，我们给出一组可能的观众投票数以产生正确结果。能够得到相同结果的观众投票数还有很多种可能。请勿将这些视为真实值，因为它们仅为示例。由于 Rachel 的评委排名为 $2^{\text{nd}}$ ，为使其合并得分最低，她需要获得最低的观众投票排名（ $4^{\text{th}}$ ），合计排名为 6。
|
||||||
|
|
||||||
|
Table 2: Example of Combining Judge and Fan Votes by Rank (Season 1, Week 4)
|
||||||
|
|
||||||
|
表 2：按排名合并评委与观众投票示例（第 1 季第 4 周）
|
||||||
|
|
||||||
|
| Contestant | Total Judges Score | Judges Score Rank | Fan Vote* | Fan Rank* | Sum of ranks |
|
||||||
|
|---|---:|---:|---:|---:|---:|
|
||||||
|
| Rachel Hunter | 25 | 2 | 1.1 million | 4 | 6 |
|
||||||
|
| Joey McIntyre | 20 | 4 | 3.7 million | 1 | 5 |
|
||||||
|
| John O’Hurley | 21 | 3 | 3.2 million | 2 | 5 |
|
||||||
|
| Kelly Monaco | 26 | 1 | 2 million | 3 | 4 |
|
||||||
|
|
||||||
|
| 选手 | 评委总分 | 评委排名 | 观众票数* | 观众排名* | 排名之和 |
|
||||||
|
|---|---:|---:|---:|---:|---:|
|
||||||
|
| Rachel Hunter | 25 | 2 | 1.1 million | 4 | 6 |
|
||||||
|
| Joey McIntyre | 20 | 4 | 3.7 million | 1 | 5 |
|
||||||
|
| John O’Hurley | 21 | 3 | 3.2 million | 2 | 5 |
|
||||||
|
| Kelly Monaco | 26 | 1 | 2 million | 3 | 4 |
|
||||||
|
|
||||||
|
* Fan vote/rank are unknown, hypothetical values chosen to produce the correct final ranks
|
||||||
|
|
||||||
|
* 观众票数/排名未知，这里为产生正确最终排名而设定的假设值
|
||||||
|
|
||||||
|
# 2. COMBINED BY PERCENT (used for season 3 through $27^{\mathrm{a}}$ )
|
||||||
|
|
||||||
|
# 2. 按百分比合并（用于第 3 季至 $27^{\mathrm{a}}$ 季）
|
||||||
|
|
||||||
|
Starting in season 3 scores were combined using percents instead of ranks. An example is shown using week 9 of season 5. In that week, Jennie Garth was eliminated. Again, we artificially created fan votes that produce total percents to correctly lead to that result. The judges' percent is computed by dividing the total judge score for the contestant by the sum of total judge scores for all 4 contestants. Based on the judges' percent, Jennie was $3^{\text{rd}}$ . However, adding the percent of the 10 million artificially created fan votes we assigned to the judges' percent she was $4^{\text{th}}$ .
|
||||||
|
|
||||||
|
从第 3 季开始，合并方式改为使用“百分比”而非“排名”。示例取自第 5 季第 9 周，当周 Jennie Garth 被淘汰。同样，我们构造了一组观众投票，使合并百分比正确产生该结果。评委百分比的计算方式为：某选手评委总分除以当周四名选手评委总分之和。按评委百分比，Jennie 排名 $3^{\text{rd}}$ 。但将我们构造的 1,000 万观众投票比例加到评委百分比后，她的合并排名变为 $4^{\text{th}}$ 。
|
||||||
|
|
||||||
|
Table 3: Example of Combining Judge and Fan Votes by Percent (Season 5, Week 9)
|
||||||
|
|
||||||
|
表 3：按百分比合并评委与观众投票示例（第 5 季第 9 周）
|
||||||
|
|
||||||
|
| Contestant | Total Judges Score | Judges Score Percent | Fan Vote* | Fan Percent* | Sum of Percent |
|
||||||
|
|---|---:|---|---:|---|---:|
|
||||||
|
| Jennie Garth | 29 | 29/117 = 24.8% | 1.1 million | 1.1/10 = 11% | 35.8 |
|
||||||
|
| Marie Osmond | 28 | 28/117 = 23.9% | 3.7 million | 3.7/10 = 37% | 60.9 |
|
||||||
|
| Mel B | 30 | 30/117 = 25.6% | 3.2 million | 3.2/10 = 32% | 57.8 |
|
||||||
|
| Helio Castroneves | 30 | 30/117 = 25.6% | 2 million | 2/10 = 20% | 45.6 |
|
||||||
|
| Total | 117 | | 10 million | | |
|
||||||
|
|
||||||
|
| 选手 | 评委总分 | 评委百分比 | 观众票数* | 观众百分比* | 百分比之和 |
|
||||||
|
|---|---:|---|---:|---|---:|
|
||||||
|
| Jennie Garth | 29 | 29/117 = 24.8% | 1.1 million | 1.1/10 = 11% | 35.8 |
|
||||||
|
| Marie Osmond | 28 | 28/117 = 23.9% | 3.7 million | 3.7/10 = 37% | 60.9 |
|
||||||
|
| Mel B | 30 | 30/117 = 25.6% | 3.2 million | 3.2/10 = 32% | 57.8 |
|
||||||
|
| Helio Castroneves | 30 | 30/117 = 25.6% | 2 million | 2/10 = 20% | 45.6 |
|
||||||
|
| Total | 117 | | 10 million | | |
|
||||||
|
|
||||||
|
* Fan vote is unknown, values hypothetical to produce the correct final standings
|
||||||
|
|
||||||
|
* 观众票数未知，这里为得到正确最终排名而设定的假设值
|
||||||
|
|
||||||
|
a The year of the return to the rank based method is not known for certain; season 28 is a reasonable assumption.
|
||||||
|
|
||||||
|
a 回归排名法的具体年份并不确定；将其视为第 28 季是合理的假设。
|
||||||
128
D_zh_en.md
Normal file
128
D_zh_en.md
Normal file
@@ -0,0 +1,128 @@
|
|||||||
|
# 2026 ICM Problem D: Managing Sports for Success
|
||||||
|
|
||||||
|
2026 ICM 题目 D：体育运营管理与成功
|
||||||
|
|
||||||
|
"The player's job is to help his team win." - Cliff Blau, baseball historian and statistician "The player's job is to make money for the owner." - all sports team owners
|
||||||
|
|
||||||
|
“球员的工作是帮助球队赢球。”——棒球史学家兼统计学家 Cliff Blau；“球员的工作是为老板赚钱。”——所有职业体育俱乐部老板
|
||||||
|
|
||||||
|
Fans tend to focus on the players on the field or court, but that is only the tip of the sports business iceberg. Sports are entertainment first and foremost. Entertainment is a profit-generating business and the players are hired mostly for that purpose. Often, fans of spectator sports ignore the financial purpose of a sport and try to focus on the game itself and its participants. However, in professional sports business, the primary goal is to make money for the owner and not necessarily win games. While these two goals may be related, since winning generates more interest in the team, other factors are involved. And for some sports teams, there are crucial moments when opportunity and risk are both high – like this year's situation for teams in the Women's National Basketball Association (WNBA), the most prominent women's professional basketball league in the United States. For many reasons (especially higher fan interest), WNBA teams are hoping to evolve from risky startup businesses into major entertainment enterprises by taking advantage of increased media attention, new team franchises, larger venues, and a new digital platform to increase revenue. The owners in that league need to use sports analytics to succeed on the court but also use financial modeling to achieve significant financial gain in the bottom line of their business's profit sheet.
|
||||||
|
|
||||||
|
球迷往往把注意力集中在场上球员身上，但这只是体育商业冰山一角。体育首先是一种娱乐。娱乐是创造利润的生意，球员的雇佣在很大程度上就是为此服务。观赛体育的粉丝常常忽视体育项目的财务目的，而试图只关注比赛本身及其参与者。然而，在职业体育商业中，首要目标是为老板赚钱，而不一定是赢球。尽管这两个目标可能相关（因为赢球会带来更多关注），但其中还涉及其他因素。对某些球队而言，还会出现“机会与风险都很高”的关键时刻——例如今年美国最具影响力的女子职业篮球联赛——女子国家篮球协会（WNBA）的多支球队所处的局面。由于多种原因（尤其是球迷兴趣提升），WNBA 球队希望借助更高的媒体关注度、新增球队特许经营、更大的比赛场馆以及新的数字平台来提高收入，从而将高风险的“初创型”业务转变为大型娱乐企业。该联盟的球队所有者既需要利用体育数据分析在赛场上取得成功，也需要运用财务建模在企业利润表的最终收益上实现显著提升。
|
||||||
|
|
||||||
|
Should players (and other employees of the team business) in a sport get paid more for their performance that produces wins or for their contributions in turning a profit for the team owner? Sometimes, a player's sport performance is directly related to profit, but not always. Some players may attract fans based more on popularity than performance. These players may generate ticket, parking, concession, and jersey revenue much more than players with higher levels of performance. Financial and sports analytics models need to connect to create good team decision making.
|
||||||
|
|
||||||
|
在一项运动中，球员（以及球队企业的其他雇员）应当因为带来胜利的赛场表现而获得更高薪酬，还是因为为球队老板创造利润的贡献而获得更高薪酬？有时，球员的竞技表现与利润直接相关，但并非总是如此。有些球员吸引球迷更多依靠人气而非表现；他们带来的门票、停车、餐饮（特许经营）以及球衣等周边收入，可能远超表现更出色的球员。要做出良好的球队决策，财务分析模型与体育数据分析模型需要彼此衔接、相互支撑。
|
||||||
|
|
||||||
|
In the emerging field of sports analytics with various kinds and amounts of performance data, there continue to be challenges to build statistics that quantify the value of player talents and performances (what statistic to measure, how to measure it, when to measure it). Some players are injured more frequently than others. How does that affect player value? Some have personalities that lead to more popularity and appeal that lead to financial gain. Context and timing matter in the sense that some players, even those with average performance, come through at important moments of the game or critical times in the season. There is a temporal element that must consider the measure of future potential of a player/employee on achieving the goal of the team. Some roles may be performance or skill-based and other roles are accomplished more by hard work and perseverance.
|
||||||
|
|
||||||
|
在体育数据分析这一新兴领域中，虽然已有多种类型、不同规模的表现数据，但要构建能够量化球员天赋与表现价值的统计指标仍面临诸多挑战（该衡量什么指标、如何衡量、何时衡量）。有些球员受伤更频繁，这会如何影响其价值？有些球员因性格特质而更具人气与吸引力，从而带来财务收益。从“语境与时点”的角度看，即便是表现中等的球员，也可能在比赛关键时刻或赛季关键阶段挺身而出。模型还需要考虑时间维度：衡量球员/雇员未来潜力对实现球队目标的作用。有些岗位更依赖表现或技能，而另一些岗位更多依靠勤奋与坚持来完成。
|
||||||
|
|
||||||
|
The player or team perceptions, popularity, timing, and marketing can play major roles, in addition to the location of the team. Teams in large markets often have different sports situations and goals than small-market teams. Those differences impact how owners achieve profit and recruit their players and employees. Can modeling help an owner establish methodologies for setting offers, negotiating, and writing contracts?
|
||||||
|
|
||||||
|
除球队所处地理位置外，外界对球员或球队的认知、人气、时机与营销也可能发挥重要作用。大市场球队的体育环境与目标往往不同于小市场球队；这些差异会影响老板如何实现盈利，以及如何招募球员和雇员。建模能否帮助球队所有者建立一套用于报价、谈判与拟定合同的系统方法？
|
||||||
|
|
||||||
|
There are many team issues that are strictly or mostly financial, just as there are issues that are mostly sports. In many cases, professional sports teams are franchises that are part of the league enterprise and often operate with additional rules and constraints set by leagues or governments on their player salaries and contracts. These are intended to make the game competition fair with some reasonable amount of competitive balance. Some professional sports have systems that regulate salaries with caps or taxes. Every season, the owner must decide how much to finance with debt versus equity and whether risks in the form of seeking better team performance with associated additional costs are worth taking. In the sports business world, conditions such as revenues, salaries, injuries, trade opportunities, taxes, fees, and interest rates change over time. Sports teams are now seen as premium assets, with values in many sports soaring far beyond historical norms due to financial and market factors such as lucrative media deals and accumulation of vast data streams and intellectual property.
|
||||||
|
|
||||||
|
球队面临的议题中，有许多完全或主要属于财务范畴，正如也有许多议题主要属于竞技体育范畴。在很多情况下，职业体育球队是联盟企业的一部分（特许经营），其运营往往还受到联盟或政府在球员薪资与合同方面制定的额外规则与约束。这些规则旨在让竞赛更公平，并保持一定程度的竞争均衡。有些职业体育项目通过工资帽或奢侈税等制度来调节薪资水平。每个赛季，老板都必须决定以负债融资还是权益融资为主，并判断为了追求更好的球队表现而承担相应的额外成本与风险是否值得。在体育商业世界中，收入、薪资、伤病、交易机会、税费与利率等条件会随时间变化。如今体育球队被视为优质资产；在不少项目中，由于高额媒体合同、海量数据流与知识产权的积累等财务与市场因素，球队估值已远超历史常态。
|
||||||
|
|
||||||
|
As a modeling group for a sports team, your ICM team can use publicly available sport and finance data for a team of your choice (the team you select must consist of at least 5 players that play cooperatively at the same time and be a member of a professional league) and build a business and management model for the team for the coming or next season.
|
||||||
|
|
||||||
|
作为某支职业运动队的建模团队，你们的 ICM 小组可以选取一支球队，并使用公开可得的体育与财务数据（所选球队必须来自职业联盟，且至少包含 5 名可同时协同上场比赛的球员），为该队即将到来或下一赛季建立一套商业与管理模型。
|
||||||
|
|
||||||
|
As was mentioned earlier as an example of how this modeling work can be extremely valuable, the WNBA is undergoing significant financial changes -- record viewership, rising franchise values, and significant player benefit expectations. Currently, negotiations and demands over the revenue-sharing agreement between teams and players are sticking points. During this coming season, team owners have an opportunity to remake and improve their business or succumb to risks that may cause them to sell or take on substantial debt. These issues create a situation where solid financial and sport modeling can make a big difference for the current and future owners of these teams. You may use a WNBA team if you care to, but you are not required to do that.
|
||||||
|
|
||||||
|
如前所述，WNBA 可以作为一个体现此类建模工作价值的典型例子：该联盟正在经历显著的财务变化——收视创纪录、特许经营球队估值上升，以及球员福利期望显著提高。目前，球队与球员之间关于收入分成协议的谈判与诉求是主要的僵持点。在即将到来的赛季里，球队所有者有机会重塑并改善其业务；也可能在风险压力下被迫出售球队或承担大量债务。这些问题使得扎实的财务与竞技建模能够对这些球队现任与未来所有者产生重大影响。你可以选择 WNBA 球队作为研究对象，但并非必须。
|
||||||
|
|
||||||
|
# Questions to consider:
|
||||||
|
|
||||||
|
# 思考问题：
|
||||||
|
|
||||||
|
Design a dynamic decision-making model that would help your team owner and general managers adjust their leverage in response to changing team performance and economic conditions. The goal is to maximize team profit and value while managing team structure and performance. The model should include priorities and actions for the management teams in both business operations and team operations, and account for systems that will help the owner make decisions through the coming season and beyond.
|
||||||
|
|
||||||
|
设计一个动态决策模型，帮助球队所有者与总经理在球队表现与经济条件变化时调整其“杠杆/议价能力”。目标是在管理球队结构与竞技表现的同时，实现球队利润与价值的最大化。模型应同时包含商业运营与球队运营两方面管理团队的优先级与行动方案，并纳入能够支撑老板在即将到来的赛季及更长周期内持续决策的机制与系统。
|
||||||
|
|
||||||
|
Based on the needs of the team and your model, develop a strategy to acquire players for next season using the standard practice for your team's league such as a draft, free agency, trades, transfer fees, or other standard practices. You may want to consider how to value a player or the
|
||||||
|
|
||||||
|
基于球队需求以及你的模型，使用该联盟的常见做法（例如选秀、自由球员签约、交易、转会费或其他标准机制）制定下一赛季的引援/补强策略。你可能需要考虑如何对球员进行估值，或如何衡量
|
||||||
|
|
||||||
|
team dynamics in terms of the profit for the team owners. Using the outcomes of your model, discuss the strengths and weaknesses of your strategy on the business.
|
||||||
|
|
||||||
|
球队内部动态对球队所有者利润的影响。基于模型输出，讨论你的策略在商业层面的优势与劣势。
|
||||||
|
|
||||||
|
There are many league-determined rulings that impact how a single team can operate, such as salary caps, number of players on a roster, schedule (number, order, location, and date of games in a season, so consequently days of rest), media contracts and rights, revenue distributions, and others. If a league is expanding the number of franchises (such as for WNBA), it is likely to impact all teams in the league. Use your model to decide how your team's strategy should change from your initial strategy during a season with league expansion. How does the location for the new team impact your model and resulting strategy? Be clear on the impact on the team owners and locations for the new team that would be particularly harmful or beneficial under an expansion.
|
||||||
|
|
||||||
|
许多由联盟制定的规则会影响单支球队的运营方式，例如：工资帽、阵容名单人数、赛程安排（赛季比赛的数量、顺序、地点与日期，从而影响休息天数）、媒体合同与转播权、收入分配等。若联盟正在扩充特许经营球队数量（如 WNBA），很可能会影响联盟内所有球队。请使用你的模型决定：在联盟扩军赛季，你的球队策略应如何在原有策略基础上调整。新球队所在地将如何影响你的模型与由此得出的策略？请明确扩军对球队所有者的影响，并指出在扩军情形下哪些新球队选址会特别有害或特别有利。
|
||||||
|
|
||||||
|
Consider one additional business decision and use your model to design the best strategy for your team. Some examples include but are not limited to:
|
||||||
|
|
||||||
|
再考虑一个额外的商业决策，并用你的模型为球队设计最佳策略。示例包括但不限于：
|
||||||
|
|
||||||
|
- Ticket sales vary greatly by the size of the stadium, time of year, popularity of the team (yours and opponent), size of the team's market, and other factors. A team may choose to maximize ticket sale revenue for each game or lower the prices to have larger attendance with the possibility to convert some of those attendees into season ticket holders. How do you determine the optimal ticket pricing strategy over a season?
|
||||||
|
|
||||||
|
门票销售会随场馆规模、季节时点、球队（己方与对手）受欢迎程度、球队市场规模等因素大幅波动。球队可以选择在每场比赛中尽可能提高门票收入，也可以通过降价提升上座率，并期望将其中一部分观众转化为季票持有者。你将如何确定一个赛季内的最优票价策略？
|
||||||
|
|
||||||
|
- The venue for the team to play its games may be rented or owned with the need to maintain, renovate, or even build a new venue. How do you balance the long-term cost of the venue when it is a short-term decision?
|
||||||
|
|
||||||
|
球队比赛场馆可能是租赁或自有资产，并可能需要维护、翻新，甚至新建。面对短期决策，你如何权衡场馆的长期成本？
|
||||||
|
|
||||||
|
- Player equity in ownership can be one strategy for subsidizing large salaries, such as revenue sharing (single season), profit participation (bonus), decision makers (as part of unions or collective bargaining), long-term equity stake (part owner), or other methods. Player equity options need to be sufficiently lucrative for a player to accept it, but not undermine the future funding options. How do you determine which players, if any, are offered equity and how much?
|
||||||
|
|
||||||
|
向球员提供所有权权益可作为补贴高薪的一种策略，例如：单赛季收入分成、利润分成（奖金）、参与决策（工会或集体谈判的一部分）、长期股权（成为部分所有者）等。权益方案需要足够吸引球员接受，同时又不能削弱球队未来融资空间。你将如何决定是否提供股权、提供给哪些球员以及提供多少？
|
||||||
|
|
||||||
|
- Media deals are a large source of revenue, fan engagement, and brand building, often producing high engagement and advertising potential. While leagues usually contract national deals, teams can sometimes broker their own local deals or streaming options. Does your team need to improve or change its media presence?
|
||||||
|
|
||||||
|
媒体合作是重要的收入来源，也能带来球迷互动与品牌建设，通常具有高参与度与广告潜力。尽管联盟通常会签订全国性转播合同，球队有时也能达成本地合作或流媒体方案。你的球队是否需要改善或调整其媒体布局与曝光方式？
|
||||||
|
|
||||||
|
- Division or conference structure, which can build or take advantage of rivalries where rival teams play more often, is generally determined by the league. Are there ways that league structures and schedules be reconfigured to increase profit for your team?
|
||||||
|
|
||||||
|
分区/分联盟结构通常由联盟决定，它可以塑造或利用“宿敌”对抗（让竞争对手更频繁交手）。是否存在重新配置联盟结构与赛程安排的方法，以提高你球队的利润？
|
||||||
|
|
||||||
|
- Determine your own issue that applies to your team or sport and use your model to help decide the issue to improve team performance or owner profit.
|
||||||
|
|
||||||
|
自行确定一个适用于你所选球队或项目的议题，并用你的模型辅助决策，以提升球队竞技表现或所有者利润。
|
||||||
|
|
||||||
|
How does your model help management adjust when a key player is injured?
|
||||||
|
|
||||||
|
当核心球员受伤时，你的模型如何帮助管理层进行调整？
|
||||||
|
|
||||||
|
Write a one- to two-page letter to your team's owner and general manager that summarizes your recommended strategy, discusses trade-offs and risks, and reflects on how your plan supports both competitive success and financial health.
|
||||||
|
|
||||||
|
撰写一封 1–2 页的信件给球队老板与总经理，概述你推荐的策略，讨论权衡与风险，并说明你的方案如何同时支持竞技成功与财务健康。
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
你的 PDF 解答总页数不超过 25 页，应包括：
|
||||||
|
|
||||||
|
- One-page Summary Sheet.
|
||||||
|
1 页摘要页（Summary Sheet）。
|
||||||
|
- Table of Contents.
|
||||||
|
目录。
|
||||||
|
- Your complete solution.
|
||||||
|
完整解答。
|
||||||
|
- One-to-Two-Page Letter.
|
||||||
|
1–2 页信件。
|
||||||
|
- References List.
|
||||||
|
参考文献列表。
|
||||||
|
- AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
AI 使用报告（如使用；不计入 25 页总页数限制。）
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
注：完整的 ICM 提交稿没有规定最低页数要求。你的全部解题内容及任何补充信息（例如：图示、示意图、计算、表格）最多可使用 25 页。允许提交部分解答。竞赛允许谨慎使用 ChatGPT 等 AI 工具，但完成本题并不依赖 AI。如你选择使用生成式 AI，必须遵循 COMAP 的 AI 使用政策；这将要求你在 PDF 解答文件末尾附加一份 AI 使用报告，该报告不计入 25 页总页数限制。
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
# 术语表
|
||||||
|
|
||||||
|
Competitive balance is how evenly matched the teams are in a league or competition.
|
||||||
|
|
||||||
|
Competitive balance（竞争均衡）指在一个联盟或赛事中，各支球队实力接近、胜负难以预测的程度。
|
||||||
|
|
||||||
|
Draft is a way for a sports league to assign new players to teams in an organized manner.
|
||||||
|
|
||||||
|
Draft（选秀）是体育联盟以组织化方式将新球员分配给各支球队的一种机制。
|
||||||
|
|
||||||
|
Free agency is a system that allows players to choose which team they will play for after their contract with a team expires.
|
||||||
|
|
||||||
|
Free agency（自由球员制度）是指球员与原球队合同到期后，可以自主选择与哪支球队签约参赛的制度。
|
||||||
152
E_zh_en.md
Normal file
152
E_zh_en.md
Normal file
@@ -0,0 +1,152 @@
|
|||||||
|
# 2026 ICM Problem E: Passive Solar Shading
|
||||||
|
|
||||||
|
2026 ICM 题目 E：被动式太阳能遮阳
|
||||||
|
|
||||||
|
# Background
|
||||||
|
|
||||||
|
# 背景
|
||||||
|
|
||||||
|
Passive solar shading has become a common addition to both housing and commercial buildings as a part of a retrofit or in new construction. It is relatively inexpensive and creates cost savings in heating and cooling. The shades are designed to block summer sun from entering a building, while allowing winter sun to not only enter the building but to warm a thermal mass that can reradiate for many hours after. Strategies such as overhangs, vegetative shading, brise-soleil systems, and high-performance glazing can reduce heat gain in buildings during higher temperatures.
|
||||||
|
|
||||||
|
被动式太阳能遮阳已成为住宅与商业建筑在改造升级或新建工程中的常见措施之一。其成本相对较低，并能在供暖与制冷方面带来费用节省。遮阳构件的设计目标是在夏季阻挡阳光进入室内，同时在冬季允许阳光进入，不仅照入室内，还可加热热质体（thermal mass），使其在随后数小时内持续再辐射释放热量。诸如外挑檐口（overhang）、植物遮阴、遮阳格栅/遮阳百叶（brise-soleil）系统以及高性能玻璃等策略，都能在高温时段降低建筑的太阳得热。
|
||||||
|
|
||||||
|
Passive solar shading is different depending on building orientation, window area distribution between the different faces of the building, and climate. It also requires the presence of an internal thermal mass that can be heated by the direct sun. This thermal mass can be concrete, stone, water, or other material that can store heat. The thermal mass not only stores heat but reduces temperature swings throughout the day.
|
||||||
|
|
||||||
|
被动式遮阳方案会因建筑朝向、各立面窗面积分布以及气候条件而不同。该策略还需要室内设置可被直射阳光加热的热质体。热质体可以是混凝土、石材、水体或其他能够储热的材料。热质体不仅能存储热量，还能减小昼夜温度波动。
|
||||||
|
|
||||||
|
These techniques use the predictable path of the sun (determined through the use of solar position calculators), materials, geometry, and natural environmental conditions to maintain comfort and reduce energy consumption. However, the typical calculations make use of the angle of the sun at solar noon on the Summer and Winter Solstices to calculate the optimal extension of a shade over a window as shown in Figure 1. This is a simplistic view of the problem, and future metrics must do better to account for change.
|
||||||
|
|
||||||
|
这些技术利用太阳可预测的运行轨迹（可通过太阳位置计算器确定），并结合材料、几何形态与自然环境条件，以维持舒适并降低能耗。然而，常见的计算方法通常仅使用夏至与冬至在当地太阳正午（solar noon）的太阳高度角，来计算窗上方遮阳构件的最优外伸长度（如图 1 所示）。这是一种较为简化的处理方式；面向未来的指标与方法需要更好地刻画变化因素。
|
||||||
|
|
||||||
|
|
||||||
|
Figure 1: Passive Solar Shading - Winter and Summer Sun on Solstices
|
||||||
|
|
||||||
|
图 1：被动式太阳能遮阳——冬至与夏至的冬季/夏季日照
|
||||||
|
|
||||||
|
|
||||||
|
# Scenario
|
||||||
|
|
||||||
|
# 情景
|
||||||
|
|
||||||
|
You have been hired by the Collective Organizations Making Astrophysical Protections (COMAP) to innovate the next generation of solar shading strategies to be implemented at the notional Sungrove University and notional Borealis University.
|
||||||
|
|
||||||
|
你受雇于 Collective Organizations Making Astrophysical Protections（COMAP），为设想的 Sungrove University 与设想的 Borealis University 创新并提出下一代太阳能遮阳策略，以供实施。
|
||||||
|
|
||||||
|
The notional Sungrove University, located in a warm, low-latitude region with high solar exposure and increasingly frequent heat waves, is planning a major transformation of its main academic quad. The campus currently suffers from excessive cooling costs and glare in the classrooms. The university leadership has decided to pursue a net-zero cooling initiative by 2040.
|
||||||
|
|
||||||
|
设想的 Sungrove University 位于温暖的低纬度地区，日照强、热浪发生频率不断上升，正计划对其主要教学四合院（main academic quad）进行重大改造。目前校园面临制冷成本过高以及教室眩光严重等问题。校方领导层已决定在 2040 年前推进“净零制冷”（net-zero cooling）行动计划。
|
||||||
|
|
||||||
|
Notably, Sungrove University is planning to retrofit its Academic Hall North. It is a two-story classroom and office building. The interior layout combines perimeter offices and classrooms with interior corridors. The building has a rectangular footprint (60m × 24m) with its long side aligned east-west as shown in Figure 2. The facade consists of double glazing and a brick veneer with an average window-to-wall ratio of $45\%$ on the south facing side and $30\%$ on the remaining sides. The building relies on mechanical cooling in the summer and hydronic heating in the winter with limited passive strategies in place. Additional features of this notional building are yours to imagine. Ensure you communicate these features in your writing to COMAP.
|
||||||
|
|
||||||
|
尤其值得注意的是，Sungrove University 计划对其 Academic Hall North 进行改造。该建筑为两层的教室与办公室综合楼，内部布局为外圈办公室与教室、内侧走廊的组合。建筑平面为矩形（60m × 24m），长边如图 2 所示沿东西方向布置。其立面由双层玻璃与砖饰面构成：南向立面的平均窗墙比约为 $45\%$ ，其余立面约为 $30\%$ 。该建筑夏季依赖机械制冷，冬季采用热水供暖（hydronic heating），目前可用的被动式策略有限。该设想建筑的其他特征可由你自行设定，并务必在提交给 COMAP 的文字中清晰说明这些设定。
|
||||||
|
|
||||||
|
|
||||||
|
Figure 2: Academic Hall North footprint
|
||||||
|
|
||||||
|
图 2：Academic Hall North 建筑平面（footprint）
|
||||||
|
|
||||||
|
Additionally, COMAP has been hired by the notional Borealis University, located at a high latitude where winter temperatures remain below freezing for months, sunlight hours are limited, and buildings experience heavy heating demands.
|
||||||
|
|
||||||
|
此外，COMAP 也受雇于设想的 Borealis University。该校位于高纬度地区，冬季气温常常连续数月低于冰点，日照时数有限，建筑供暖需求很高。
|
||||||
|
|
||||||
|
Sungrove University and Borealis University are also both planning a new student union that will serve as the hub of university activities. They have each mandated that their new student union building relies heavily on passive solar shading rather than mechanical cooling systems. The Universities want their student union building to serve as a prototype for future developments, meaning that their passive solar strategy design must perform well not only today, but under projected climate conditions well into the future.
|
||||||
|
|
||||||
|
Sungrove University 与 Borealis University 还都计划新建一座学生中心（student union），作为校园活动的枢纽。两校均要求新建学生中心应主要依赖被动式太阳能遮阳，而非机械制冷系统。校方希望该学生中心能成为未来开发建设的原型（prototype），这意味着其被动式太阳能策略不仅要在当下有效，也必须在未来气候情景预测条件下长期保持良好性能。
|
||||||
|
|
||||||
|
Beyond the standard approach to shading as outlined in the Background, to assist these notional universities, you should extend your ideas to include:
|
||||||
|
|
||||||
|
在背景部分所述的标准遮阳方法之外，为协助这些设想大学，你需要将构思扩展到包括：
|
||||||
|
|
||||||
|
- Shading needs throughout the day rather than just at solar noon.
|
||||||
|
- Windows of different sizes and shapes.
|
||||||
|
- Windows that do not face exactly south/north (depending on the hemisphere).
|
||||||
|
- Shades of different styles and materials that would match the architecture of the building.
|
||||||
|
|
||||||
|
- 不仅考虑太阳正午（solar noon），还要考虑全天各时段的遮阳需求。
|
||||||
|
- 不同尺寸与形状的窗。
|
||||||
|
- 并非严格朝南/朝北（取决于所在半球）的窗。
|
||||||
|
- 与建筑风格相匹配的不同样式与材料的遮阳构件。
|
||||||
|
|
||||||
|
As with any new strategy or model, you will not only need to describe your approach but also explain the advantages that your proposal holds over the previous standard. COMAP needs to know how your passive solar shading strategies can more effectively reduce heat gain in campus buildings during the summer while still admitting beneficial winter sun.
|
||||||
|
|
||||||
|
与任何新策略或新模型一样，你不仅需要描述你的方案，还需要说明其相较于既有标准方法的优势。COMAP 需要了解：你的被动式太阳能遮阳策略如何在夏季更有效地减少校园建筑的太阳得热，同时仍能引入冬季有益的日照。
|
||||||
|
|
||||||
|
# Requirements
|
||||||
|
|
||||||
|
# 要求
|
||||||
|
|
||||||
|
Your team has been asked by COMAP to provide a model-based feasibility analysis that determines how Sungrove University can reduce its academic year cooling load with passive solar design in the retrofit of buildings on campus. To do so, design a retrofit for Sungrove University's Academic Hall North that optimizes heating and cooling throughout the academic year. What passive solar strategies and building features would you use, and how would you evaluate their performance?
|
||||||
|
|
||||||
|
COMAP 要求你的团队提供一份基于模型的可行性分析，以确定 Sungrove University 如何通过在校园建筑改造中采用被动式太阳能设计，降低学年期间的制冷负荷。为此，请为 Sungrove University 的 Academic Hall North 设计一套改造方案，使其在整个学年内的供暖与制冷达到优化。你将采用哪些被动式太阳能策略与建筑特征？又将如何评估其性能？
|
||||||
|
|
||||||
|
Borealis University has a building with a similar design to Sungrove University's Academic Hall North. How can extending your work for Sungrove University to include the crucial importance of the effective use of a thermal mass provide Borealis University with a plan to use passive solar shading? You may want to consider building geometry, material selection, glazing positioning, internal thermal mass, or other aspects to maximize winter heat gain while avoiding overheating in the warmer months.
|
||||||
|
|
||||||
|
Borealis University 有一栋与 Sungrove University 的 Academic Hall North 设计相近的建筑。若将你在 Sungrove University 的工作扩展到强调并纳入“有效利用热质体”的关键作用，如何为 Borealis University 提供一套使用被动式太阳能遮阳的方案？你可以考虑建筑几何形态、材料选择、玻璃布置、室内热质体设置等因素，以在最大化冬季得热的同时，避免在较温暖月份出现过热。
|
||||||
|
|
||||||
|
The retrofit design models at both Sungrove and Borealis Universities are helpful for only those notional sites. Adapt your model and discuss the design considerations for other locations including the different heating and cooling needs at places that might have similar latitudes.
|
||||||
|
|
||||||
|
针对 Sungrove 与 Borealis 两校的改造设计模型仅适用于这两个设想场址。请对模型进行调整，并讨论适用于其他地点的设计考量，包括：在纬度相近但供暖与制冷需求不同的地区，你将如何改动与应用你的模型。
|
||||||
|
|
||||||
|
Design a passive solar shading strategy for the new student union building at either Sungrove University or Borealis University that keeps the building temperate. Describe the strategies, building features, and modeling approaches you would use to evaluate performance over time. You may wish to address some of the following in your analysis:
|
||||||
|
|
||||||
|
为 Sungrove University 或 Borealis University（二者择一）的新学生中心设计一套被动式太阳能遮阳策略，使建筑保持适宜的室内热环境。请描述你将采用的策略、建筑特征与建模方法，以及如何随时间评估其性能。你的分析可考虑（但不限于）以下方面：
|
||||||
|
|
||||||
|
- Predicting solar heat gain
|
||||||
|
- Estimating heating and/or cooling load reductions
|
||||||
|
- Accounting for seasonal variations
|
||||||
|
- Evaluating the tradeoffs between daylighting needs and shading effectiveness
|
||||||
|
|
||||||
|
- 预测太阳得热
|
||||||
|
- 估算供暖与/或制冷负荷的降低幅度
|
||||||
|
- 纳入季节变化因素
|
||||||
|
- 评估采光需求与遮阳效果之间的权衡
|
||||||
|
|
||||||
|
Write a one-to-two-page letter to either Sungrove University or Borealis University (not both) outlining the steps they should take to include passive solar shading in both their retrofit and new building plans.
|
||||||
|
|
||||||
|
给 Sungrove University 或 Borealis University（仅选其一）撰写一封 1–2 页的信件，概述他们在既有建筑改造与新建项目中纳入被动式太阳能遮阳应采取的步骤。
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
你的 PDF 解答总页数不超过 25 页，应包括：
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One-to-Two-Page Letter.
|
||||||
|
- References List.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
1 页摘要页（Summary Sheet）。
|
||||||
|
目录。
|
||||||
|
- 完整解答。
|
||||||
|
1–2 页信件。
|
||||||
|
- 参考文献列表。
|
||||||
|
AI 使用报告（如使用；不计入 25 页总页数限制。）
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
注：完整的 ICM 提交稿没有规定最低页数要求。你的全部解题内容及任何补充信息（例如：图示、示意图、计算、表格）最多可使用 25 页。允许提交部分解答。竞赛允许谨慎使用 ChatGPT 等 AI 工具，但完成本题并不依赖 AI。如你选择使用生成式 AI，必须遵循 COMAP 的 AI 使用政策；这将要求你在 PDF 解答文件末尾附加一份 AI 使用报告，该报告不计入 25 页总页数限制。
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
# 术语表
|
||||||
|
|
||||||
|
Solar noon is the moment during the day when the Sun is at its highest point in the sky for a given location.
|
||||||
|
|
||||||
|
Solar noon（太阳正午）是指在某一地点的一天中，太阳在天空中达到最高点的时刻。
|
||||||
|
|
||||||
|
Winter Solstice is the day with the least daylight of the year, caused by Earth's tilt.
|
||||||
|
|
||||||
|
Winter Solstice（冬至）是指一年中白昼最短的一天，其成因与地轴倾角有关。
|
||||||
|
|
||||||
|
Summer Solstice is the day with the most daylight of the year, caused by Earth's tilt.
|
||||||
|
|
||||||
|
Summer Solstice（夏至）是指一年中白昼最长的一天，其成因与地轴倾角有关。
|
||||||
|
|
||||||
|
Notional means theoretical or fictitious. The universities in this problem are not real, but only theoretical case studies.
|
||||||
|
|
||||||
|
Notional 意为“设想的/虚拟的/理论上的”。本题中的两所大学并非真实存在，仅为理论案例研究。
|
||||||
|
|
||||||
|
Net-zero cooling means providing cooling without adding greenhouse gases to the atmosphere.
|
||||||
|
|
||||||
|
Net-zero cooling（净零制冷）是指在实现制冷的同时，不向大气增加温室气体排放（实现净零增量排放）。
|
||||||
75
F_zh_en.md
Normal file
75
F_zh_en.md
Normal file
@@ -0,0 +1,75 @@
|
|||||||
|
# 2026 ICM
|
||||||
|
|
||||||
|
2026 ICM
|
||||||
|
|
||||||
|
# Problem F: To Gen-AI, or Not To Gen-AI (or how to Gen-AI)? That is the Question!
|
||||||
|
|
||||||
|
题目 F：用还是不用生成式 AI（或如何使用生成式 AI）？这是个问题！
|
||||||
|
|
||||||
|
In just a few years, Generative Artificial Intelligence (Gen-AI) has gone from a tool of limited capacity, only used by a few early-adopters to a powerful and inescapable resource embedded in our daily lives. Over time, research suggests that Gen-AI could impact the future of work. For example, in some fields, Gen-AI may replace humans (or heavily reduce the human workload), while other fields might not be heavily impacted or might even grow.
|
||||||
|
|
||||||
|
仅在短短数年间，生成式人工智能（Gen-AI）就从一种能力有限、仅被少数先行者使用的工具，演变为嵌入日常生活的强大且难以回避的资源。随着时间推移，研究表明 Gen-AI 可能影响未来的工作形态。例如，在某些领域，Gen-AI 可能取代人类（或大幅减少人类工作量）；而在另一些领域，影响可能不大，甚至可能促进该领域增长。
|
||||||
|
|
||||||
|
In this question, you will explore how post-secondary educational institutions, of various types, should best prepare their future graduates in light of this new technology. Specifically, you are asking do to the following.
|
||||||
|
|
||||||
|
在本题中，你将探讨不同类型的高等教育后阶段（post-secondary）机构应如何在这一新技术背景下更好地培养未来毕业生。具体而言，你需要完成以下任务。
|
||||||
|
|
||||||
|
- Choose three careers, one from each of the following categories:
|
||||||
|
|
||||||
|
- 选择三种职业，每类各选一种：
|
||||||
|
|
||||||
|
STEM career: people in this career often have at least a four year university degree in the sciences, engineering, or mathematics;
|
||||||
|
Trade career: people in this career often have training from a trade school and/or an apprenticeship program, such as chef, plumber, and electrician;
|
||||||
|
○ Arts career: people in this career often have studied at an arts school, conservatory, or cultural center, such as musician, dancer, or painter.
|
||||||
|
|
||||||
|
STEM 职业：从业者通常至少拥有科学、工程或数学方向的四年制大学学位；
|
||||||
|
技工/职业技能类职业（Trade career）：从业者通常接受职业学校培训和/或学徒制训练，例如厨师、水管工、电工；
|
||||||
|
○ 艺术类职业（Arts career）：从业者通常曾在艺术学校、音乐学院/艺术学院（conservatory）或文化中心学习，例如音乐家、舞者或画家。
|
||||||
|
|
||||||
|
- Design a data-informed model to explore the future of each of your three chosen professions, given the current trajectory and expected impacts of Gen-AI. Be sure to identify your data sources as well as your reasoning behind any drivers that you expect to change this profession as a result of Gen-AI. Note: you may leverage existing research on the future of work, but be sure to cite your sources and explain how you are using the established research to inform your analysis.
|
||||||
|
|
||||||
|
- 设计一个数据驱动（data-informed）的模型，基于 Gen-AI 的当前发展轨迹及其预期影响，分析你所选三种职业的未来走向。请明确你的数据来源，并阐述你所认为会因 Gen-AI 而改变该职业的关键驱动因素及其理由。注：你可以利用关于未来工作的现有研究，但务必规范引用来源，并解释你如何使用这些既有研究来支撑你的分析。
|
||||||
|
|
||||||
|
- Identify a specific post-secondary institution and program of study for each career you are analyzing (one at a university, one at a trade school, one at an arts school), and focus your recommendations accordingly. In other words, you should have three sets of recommendations that address the following question: Based on your analysis, how would you advise the leaders of each of these institutions to address Gen-AI in the programs specific to the careers you are analyzing?
|
||||||
|
|
||||||
|
- 针对你分析的每一种职业，分别选定一个具体的高等教育后阶段机构及其对应的学习项目/专业（一个在大学、一个在职业学校、一个在艺术学校），并据此聚焦你的建议。换言之，你需要提出三套建议，回答以下问题：基于你的分析，你会如何建议这些机构的领导者在与你所分析职业相关的培养项目中应对 Gen-AI？
|
||||||
|
|
||||||
|
Below are just some thoughts you may wish to consider; teams should not attempt to address all of these ideas, but should use these as inspiration that will lead to a cogent and thorough analysis that should vary team to team.
|
||||||
|
|
||||||
|
以下仅为一些可供考虑的思路；各队不应试图面面俱到，而应将其作为启发，形成连贯且充分的分析（不同队伍的分析应有所差异）。
|
||||||
|
|
||||||
|
○ Should the program of study grow or shrink (graduate more or fewer people) as a result of changes in the career due to Gen-AI? If the field should grow, how might the institution recruit more people; and if the field should shrink, are there other programs in the school that should grow to absorb the people who used to study in this program?
|
||||||
|
|
||||||
|
○ 随着 Gen-AI 导致职业发生变化，相关培养项目规模应扩大还是缩小（毕业人数更多或更少）？若该领域应扩大，学校如何招募更多学生；若该领域应缩小，学校是否应扩大其他项目以吸纳原本会就读该项目的学生？
|
||||||
|
|
||||||
|
○ What should these three different programs of study teach about Gen-AI? Many post-secondary institutions of learning have asked this question and are still developing their response. While some institutions have outright banned the use of AI on any assignments, others have brought the use of AI to the forefront of their curriculum. Some schools aim to produce experts who can contribute to the leading edge of the technological field, while some focus on graduating students in non-technical fields who are fluent users of the technology. Some institutions encourage their students to think about all the ways they can apply this new technology, and some schools challenge students to carefully weigh the benefits and costs of using AI, given the requisite energy usage, water demands, and risk of insufficient (often missing or incorrect) attribution to the original creators of ideas or content. For the three programs of study at the three institutions you've selected, what do you recommend to best support the employability of their graduates? Be sure to support your recommendations with the results of a mathematical model.
|
||||||
|
|
||||||
|
○ 这三类不同的培养项目应当教授哪些与 Gen-AI 相关的内容？许多高等教育后阶段机构都提出了这一问题，并仍在形成回应。有些机构直接禁止在任何作业中使用 AI；另一些则将 AI 使用置于课程体系的核心。有的学校旨在培养能够推动技术前沿发展的专家；也有学校侧重于培养非技术领域但能熟练使用该技术的毕业生。有些机构鼓励学生思考可将新技术应用到哪些方面；也有学校要求学生在考虑 AI 所需能耗、水资源需求，以及对原创思想或内容创作者的署名/归因不足（常见为缺失或不准确）风险的前提下，审慎权衡使用 AI 的收益与成本。针对你所选三所机构的三项培养项目，你建议如何做才能最有效地提升毕业生的可就业性？务必用数学模型的结果来支撑你的建议。
|
||||||
|
|
||||||
|
While this problem poses the question through the context of employability of graduates in a world where Gen-AI is ubiquitous, perhaps employment demands are not the only way to measure the success of the institutional policies you are proposing. What other factors do you believe should be considered, and how do your models and recommendations change when you consider these other factors?
|
||||||
|
|
||||||
|
尽管本题以“在 Gen-AI 无处不在的世界中毕业生的可就业性”为背景提出问题，但就业需求也许并非衡量你所提出机构政策成败的唯一标准。你认为还应考虑哪些因素？当你将这些因素纳入考量时，你的模型与建议将如何变化？
|
||||||
|
|
||||||
|
If you believe your specific recommendations can be generalized beyond one institution and/or beyond one program, be sure to explain the extent of the generalization and justify this.
|
||||||
|
|
||||||
|
如果你认为你的具体建议可以推广到某一所机构之外和/或某一个项目之外，请明确说明可推广的范围，并给出论证理由。
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
你的 PDF 解答总页数不超过 25 页，应包括：
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
- References list.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
1 页摘要页（Summary Sheet）。
|
||||||
|
目录。
|
||||||
|
- 完整解答。
|
||||||
|
- 参考文献列表。
|
||||||
|
AI 使用报告（如使用；不计入 25 页总页数限制。）
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
注：完整的 ICM 提交稿没有规定最低页数要求。你的全部解题内容及任何补充信息（例如：图示、示意图、计算、表格）最多可使用 25 页。允许提交部分解答。竞赛允许谨慎使用 ChatGPT 等 AI 工具，但完成本题并不依赖 AI。如你选择使用生成式 AI，必须遵循 COMAP 的 AI 使用政策；这将要求你在 PDF 解答文件末尾附加一份 AI 使用报告，该报告不计入 25 页总页数限制。
|
||||||
BIN
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66
en_A.md
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66
en_A.md
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@@ -0,0 +1,66 @@
|
|||||||
|
# 2026 MCM
|
||||||
|
|
||||||
|
# Problem A: Modeling Smartphone Battery Drain
|
||||||
|
|
||||||
|
|
||||||
|
25%
|
||||||
|
|
||||||
|
|
||||||
|
50%
|
||||||
|
|
||||||
|
|
||||||
|
75%
|
||||||
|
|
||||||
|
|
||||||
|
100%
|
||||||
|
|
||||||
|
Smartphones are indispensable tools in modern life, yet their battery behavior often seems unpredictable. On some days a phone may last the whole day; on other days it drains rapidly before lunch. Although some users attribute this to "heavy use," the true drivers of battery depletion are more complex. Power consumption depends on the interplay of screen size and brightness, processor load, network activity, and background applications that continue drawing energy even when the device appears idle. Environmental conditions such as temperature further complicate matters: some batteries lose effective capacity in cold weather and may overheat under sustained heavy use. A battery's behavior is also influenced by its history and how it has been charged during its lifetime.
|
||||||
|
|
||||||
|
Your task is to develop a continuous-time mathematical model of a smartphone's battery that returns the state of charge (SOC) as a function of time under realistic usage conditions. This will be used to predict the remaining time-to-empty under different conditions. You should assume that the phone has a lithium-ion battery.
|
||||||
|
|
||||||
|
# Requirements:
|
||||||
|
|
||||||
|
1. Continuous-Time Model: Develop a model to represent the state of charge using a continuous-time equation or system of equations. You may want to begin with the simplest reasonable description of battery drain and then extend it to incorporate additional contributors such as screen usage, processor load, network connections, GPS usage, and other background tasks.
|
||||||
|
|
||||||
|
Data as support, not substitute: You may collect or use data for parameter estimation and validation. If open datasets are limited, you may use published measurements or specifications (with proper citation), provided parameters are clearly justified and validated for plausibility. However, projects based solely on discrete curve fitting, timestep regression, or black-box machine learning without an explicit continuous-time model will not satisfy this problem's requirements. All data used must be well documented and freely available, and the data must be free for use under an open license.
|
||||||
|
|
||||||
|
2. Time-to-Empty predictions: Use your model to compute or approximate the time-to-empty under various initial charge levels and usage scenarios. Compare predictions to observed or plausible behavior, quantify uncertainty, and identify where the model performs well or poorly.
|
||||||
|
|
||||||
|
○ Show how your model explains differences in these outcomes and identify the specific drivers of rapid battery drain in each case.
|
||||||
|
- Which activities or conditions produce the greatest reductions in battery life? Which ones change the model surprisingly little?
|
||||||
|
|
||||||
|
3. Sensitivity and Assumptions: Examine how your predictions vary after making changes in your modeling assumptions, parameter values, and fluctuations in usage patterns.
|
||||||
|
|
||||||
|
4. Recommendations: Translate your findings into practical recommendations for a cellphone user. For example, which user behaviors—such as reducing brightness, disabling background tasks, or switching network modes—yield the largest improvements in battery life? How might an operating system implement more effective power-saving strategies based on insights from your model? Consider how battery aging reduces effective capacity or how your modeling framework could generalize to other portable devices.
|
||||||
|
|
||||||
|
# Your report should present:
|
||||||
|
|
||||||
|
- A clear description of your model and governing equations.
|
||||||
|
- The assumptions and rationale behind your design choices.
|
||||||
|
Parameter estimation methods and validation results.
|
||||||
|
- A discussion of strengths, limitations, and possible extensions.
|
||||||
|
- An executive-style summary highlighting main results, insights, and recommendations.
|
||||||
|
|
||||||
|
Important: Your model must be grounded in clearly defined physical or mechanical reasoning; discrete curve fitting or other mathematical forms that are disconnected from an explicit continuous-time description of battery behavior will not satisfy the requirements. Projects that rely solely on discrete curve fitting or statistical regression without a clearly formulated continuous-time model will not satisfy the requirements of this problem.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
In-text Citations and A Reference List.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
Smartphone: is a mobile device that combines the functionality of a traditional cell phone with advanced computing capabilities.
|
||||||
|
|
||||||
|
Power Consumption: the rate at which a device uses electrical energy from its battery or power source.
|
||||||
|
|
||||||
|
Processor Load: the actual amount of work being done by the processor at a given moment.
|
||||||
|
|
||||||
|
State of Charge (SOC): a measure of how much energy remains in a battery compared to its full capacity, expressed as a percentage.
|
||||||
|
|
||||||
|
Time-to-Empty: the estimated amount of time remaining before a battery is completely discharged.
|
||||||
59
en_B.md
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59
en_B.md
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@@ -0,0 +1,59 @@
|
|||||||
|
# 2026 MCM
|
||||||
|
|
||||||
|
# Problem B: Creating a Moon Colony Using a Space Elevator System
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
Imagine a future where it's possible for anyone to visit space by taking a leisurely and scenic ride from the Equator to Earth's orbit and then catching a routine, safe, and inexpensive rocket flight to the Moon, Mars, or beyond. In this future, we could build lush, green, and beautiful space habitats with artificial gravity, where people would vacation, work, or even live. These habitats would alleviate pressure on Earth's delicate, overworked, and fragile ecosystems. The technology to enable these events would provide humankind with limitless, safe, routine, environmentally friendly, efficient, and global access to space. To achieve these goals, some people envision a Space Elevator System, powered by electricity, offering a scalable infrastructure for interplanetary logistics, commerce, and exploration.
|
||||||
|
|
||||||
|
At its final operating configuration, the Space Elevator System would comprise three Galactic Harbours, ideally separated by 120 degrees around the equator. Each Galactic Harbour would include a single Earth port with two $100,000\mathrm{km}$ -long tethers connected to two apex anchors, with multiple space elevators operating together, each capable of lifting massive payloads daily from Earth to geosynchronous orbit (GEO) and beyond to the apex anchor where they can be loaded on a rocket and delivered anywhere using much less fuel.
|
||||||
|
|
||||||
|
The Moon Colony Management (MCM) Agency is preparing to build a Moon Colony with an estimated 100,000 people beginning in the year 2050, after completion of the Space Elevator System. It is estimated that the Moon Colony will need about 100 million metric tons of materials. Additionally, water and supplies will routinely need to be sent to sustain the Moon's population once the colony is complete. To get to the Moon, the Galactic Harbour must send material in two steps: first, from the Earth port to the apex anchor via a space elevator, and second, from the apex anchor to the Moon Colony via a rocket. The MCM Agency anticipates that the Galactic Harbour will provide an advanced lift system capable of moving 179,000 metric tons every year, while generating no atmospheric pollution.
|
||||||
|
|
||||||
|
The agency is also considering using traditional rockets to supply material for construction and supplies to the Moon Colony. The Earth current has ten rocket launch sites: Alaska, California, Texas, Florida, and Virginia (United States), Kazakhstan, French Guiana, Satish Dhawan Space Centre (India), Taiyuan Satellite Launch Center (China), and Mahia Peninsula (New Zealand).
|
||||||
|
|
||||||
|
A rocket would require a single step from the rocket launch site on Earth to the Moon Colony. By 2050 it is estimated that rockets will be able to carry 100-150 metric tons of payload to the Moon using advanced Falcon Heavy launches. You may assume perfect conditions for both the Galactic Harbour system (e.g., no swaying of the tether) and rocket launches (e.g., no failed launches). You should consider the cost and timeline to deliver the materials from the surface of the Earth to the Moon Colony site for the different scenarios.
|
||||||
|
|
||||||
|
# Your Task:
|
||||||
|
|
||||||
|
Your task is to utilize a mathematical model to determine the cost and associated timeline in order to transport material to build a 100,000 person Moon Colony starting in 2050. You will need to compare the Modern-Day Space Elevator System's three Galactic Harbours to traditional rockets launched from selected rocket bases.
|
||||||
|
|
||||||
|
# Your model should include:
|
||||||
|
|
||||||
|
1. Consideration of three different scenarios for how the 100 million metric tons of materials will be delivered to build the 100,000-person Moon Colony;
|
||||||
|
|
||||||
|
a. using the Space Elevator System's three Galactic Harbor's alone,
|
||||||
|
b. traditional rocket launches from existing bases alone (you may choose which facilities to use), or,
|
||||||
|
c. some combination of the two methods.
|
||||||
|
|
||||||
|
2. To what extent does your solution(s) change if the transportation systems are not in perfect working order (e.g, swaying of the tether, rockets fail, elevators break, etc.).?
|
||||||
|
3. Investigate the water needs for a one-year period once the 100,000-person Moon Colony is fully operational. Use your delivery model to understand the additional cost and timeline needed to ensure the colony has sufficient water for one full year after the Moon Colony is inhabited.
|
||||||
|
4. Discuss the impact on the Earth's environment for achieving the 100,000-person Moon Colony under the different scenarios. How would you adjust your model to minimize the environmental impact?
|
||||||
|
5. Write a one-page letter recommending a course of action to the fictional MCM Agency to build and sustain a 100,000-person Moon Colony.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One-page letter to MCM Agency
|
||||||
|
References list.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
Space Elevator System is comprised of three Galactic Harbours plus additional support facilities.
|
||||||
|
|
||||||
|
Galactic Harbour is comprised of two apex anchors each connected by two tethers to a single Earth Port.
|
||||||
|
|
||||||
|
Earth Port is the location on the Earth that provides surface support for the Galactic Harbour.
|
||||||
|
|
||||||
|
Tethers are $100,000\mathrm{km}$ long graphene material that links the Earth port and apex anchors in the Space Elevator System.
|
||||||
|
|
||||||
|
Apex Anchor is the counterweight in space at the end of the $100,000\mathrm{km}$ tether.
|
||||||
|
|
||||||
|
Geosynchronous orbit (GEO) is approximately $35,786\mathrm{km}$ above the surface of the Earth where the orbital period to circle Earth is 24 hours, matching Earth's rotation so it stays over the same longitude each day.
|
||||||
|
|
||||||
|
Moon Colony is a habitat on the moon with the capacity to support 100,000-persons.
|
||||||
95
en_C.md
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95
en_C.md
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@@ -0,0 +1,95 @@
|
|||||||
|
# 2026 MCM Problem C: Data With The Stars
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
Dancing with the Stars (DWTS) is the American version of an international television franchise based on the British show "Strictly Come Dancing" ("Come Dancing" originally). Versions of the show have appeared in Albania, Argentina, Australia, China, France, India, and many other countries. The U.S. version, the focus of this problem, has completed 34 seasons.
|
||||||
|
|
||||||
|
Celebrities are partnered with professional dancers and then perform dances each week. A panel of expert judges scores each couple's dance, and fans vote (by phone or online) for their favorite couple that week. Fans can vote once or multiple times up to a limit announced each week. Further, fans vote for the star they wish to keep, but cannot vote to eliminate a star. The judge and fan votes are combined in order to determine which couple to eliminate (the lowest combined score) that week. Three (in some seasons more) couples reach the finals and in the week of the finals the combined scores from fans and judges are used to rank them from $1^{\text{st}}$ to $3^{\text{rd}}$ (or $4^{\text{th}}$ , $5^{\text{th}}$ ).
|
||||||
|
|
||||||
|
There are many possible methods of combining fan votes and judge scores. In the first two seasons of the U.S. show, the combination was based on ranks. Season 2 concerns (due to celebrity contestant Jerry Rice who was a finalist despite very low judge scores) led to a modification to use percentages instead of ranks. Examples of these two approaches are provided in the Appendix.
|
||||||
|
|
||||||
|
In season 27, another "controversy" occurred when celebrity contestant Bobby Bones won despite consistently low judges scores. In response, starting in season 28 a slight modification to the elimination process was made. The bottom two contestants were identified using the combined judge scores and fan votes, and then during the live show the judges voted to select which of these two to eliminate. Around this same season, the producers also returned to using the method of ranks to combine judges scores with fan votes as in seasons one and two. The exact season this change occurred is not known, but it is reasonable to assume it was season 28.
|
||||||
|
|
||||||
|
Judge scores are meant to reflect which dancers are technically better, although there is some subjectivity in what makes a dance better. Fan votes are likely much more subjective, influenced by the quality of the dance, but also the popularity and charisma of the celebrity. Show producers might actually prefer, to some extent, conflicts in opinions and votes as such occurrences boost fan interest and excitement.
|
||||||
|
|
||||||
|
Data with judges scores and contestant information is provided and described below. You may choose to include additional information or other data at your discretion, but you must completely document the sources. Use the data to:
|
||||||
|
|
||||||
|
- Develop a mathematical model (or models) to produce estimated fan votes (which are unknown and a closely guarded secret) for each contestant for the weeks they competed.
|
||||||
|
|
||||||
|
- Does your model correctly estimate fan votes that lead to results consistent with who was eliminated each week? Provide measures of the consistency.
|
||||||
|
- How much certainty is there in the fan vote totals you produced, and is that certainty always the same for each contestant/week? Provide measures of your certainty for the estimates.
|
||||||
|
|
||||||
|
- Use your fan vote estimates with the rest of the data to:
|
||||||
|
|
||||||
|
○ Compare and contrast the results produced by the two approaches used by the show to combine judge and fan votes (i.e. rank and percentage) across seasons (i.e. apply both approaches to each season). If differences in outcomes exist, does one method seem to favor fan votes more than the other?
|
||||||
|
○ Examine the two voting methods applied to specific celebrities where there was “controversy”, meaning differences between judges and fans. Would the choice of method to combine judge scores and fan votes have led to the same result for each of these contestants? How would including the additional approach of having judges choose which of the bottom two couples to eliminate each week impact the results? Some examples you might consider (there may also be others you identified):
|
||||||
|
|
||||||
|
- season 2 - Jerry Rice, runner up despite the lowest judges scores in 5 weeks.
|
||||||
|
- season 4 - Billy Ray Cyrus was $5^{\text{th}}$ despite last place judge scores in 6 weeks.
|
||||||
|
- season 11 - Bristol Palin was $3^{\text{rd}}$ with the lowest judge scores 12 times.
|
||||||
|
- season 27 - Bobby Bones won the despite consistently low judges scores
|
||||||
|
|
||||||
|
- Based on your analysis, which of the two methods would you recommend using for future seasons and why? Would you suggest including the additional approach of judges choosing from the bottom two couples?
|
||||||
|
|
||||||
|
- Use the data including your fan vote estimates to develop a model that analyzes the impact of various pro dancers as well as characteristics for the celebrities available in the data (age, industry, etc). How much do such things impact how well a celebrity will do in the competition? Do they impact judges scores and fan votes in the same way?
|
||||||
|
- Propose another system using fan votes and judge scores each week that you believe is more "fair" (or "better" in some other way such as making the show more exciting for the fans). Provide support for why your approach should be adopted by the show producers.
|
||||||
|
- Produce a report of no more than 25 pages with your findings and include a one- to two-page memo summarizing your results with advice for producers of DWTS on the impact of how judge and fan votes are combined with recommendations for how to do so in future seasons.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One- to two-page memo.
|
||||||
|
- References list.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
Data File: 2026_MCM_Problem_C_Data.csv – contestant information, results, and judges scores by week for seasons 1 – 34. The data description is provided in Table 1.
|
||||||
|
|
||||||
|
Table 1: Data Description for 2026_MCM_Problem_C_Data.csv
|
||||||
|
|
||||||
|
<table><tr><td>Variables</td><td>Explanation</td><td>Example</td></tr><tr><td>celebrity_name</td><td>Name of celebrity contestant (Star)</td><td>Jerry Rice, Mark Cuban, ...</td></tr><tr><td>ballroompartner</td><td>Name of professional dancer partner</td><td>Cheryl Burke, Derek Hough, ...</td></tr><tr><td>celebrity_industry</td><td>Star profession category</td><td>Athlete, Model, ...</td></tr><tr><td>celebrity_homestate</td><td>Star home state (if from U.S.)</td><td>Ohio, Maine, ...</td></tr><tr><td>celebrity_homecountry/region</td><td>Star home country/region</td><td>United States, England, ...</td></tr><tr><td>celebrity_age during season</td><td>Age of the star in the season</td><td>32, 29, ...</td></tr><tr><td>season</td><td>Season of the show</td><td>1, 2, 3, ..., 32</td></tr><tr><td>results</td><td>Season results for the start</td><td>1st Place, Eliminated Week 2, ...</td></tr><tr><td>placement</td><td>Final place for the season (1 best)</td><td>1, 2, 3, ...</td></tr><tr><td>weekXjudgeY_score</td><td>Score from judge Y in week X</td><td>1, 2, 3, ...</td></tr></table>
|
||||||
|
|
||||||
|
# Notes on the data:
|
||||||
|
|
||||||
|
1. Judges scores for each dance are from 1 (low) to 10 (high).
|
||||||
|
|
||||||
|
a. In some weeks the score reported includes a decimal (e.g. 8.5) because each celebrity performed more than one dance and the scores from each are averaged.
|
||||||
|
b. In some weeks, bonus points were awarded (dance offs etc); they are spread evenly across judge/dance scores.
|
||||||
|
c. Team dance scores were averaged with scores for each individual team member.
|
||||||
|
|
||||||
|
2. Judges are listed in the order they scored dances; thus "Judge Y" may not be the same judge from week to week, or season to season.
|
||||||
|
|
||||||
|
3. The number of celebrities is not the same across the seasons, nor is the number of weeks the show ran.
|
||||||
|
4. Season 15 was the only season to feature an all-star cast of returning celebrities.
|
||||||
|
5. There are occasionally weeks when no celebrity was eliminated, and others where more than one was eliminated.
|
||||||
|
6. $N / A$ values occur in the data set for
|
||||||
|
a. the $4^{th}$ judge score if there is not $4^{th}$ judge for that week (usually there are 3) and
|
||||||
|
b. in weeks that the show did not run in a season (for example, season 1 lasted 6 weeks so $N / A$ values are recorded for weeks 7 thru 11).
|
||||||
|
7. A 0 score is recorded for celebrities who are eliminated. For example, in Season 1 the first celebrity eliminated was Trista Sutter at the end of the Week 2 show. She thus has scores of 0 for the rest of the season (week 3 through week 6).
|
||||||
|
|
||||||
|
# Appendix: Examples of Voting Schemes
|
||||||
|
|
||||||
|
# 1. COMBINED BY RANK (used in seasons 1, 2, and $28^{\mathrm{a}}$ - 34)
|
||||||
|
|
||||||
|
In seasons 1 and 2 judges and fan votes were combined by rank. For example, in season 1, week 4 there were four remaining contestants. Rachel Hunter was eliminated meaning she received the lowest combined rank. In Table 2 the judges scores and ranks are shown, and we created one possible set of fan votes that would produce the correct result. There are many possible values for fan votes that would also give the same results. You should not use these as actual values as this is just one example. Since Rachel was ranked $2^{\text{nd}}$ by judges, in order to finish with the lowest combined score, she has the lowest fan vote ( $4^{\text{th}}$ place) for a total rank of 6.
|
||||||
|
|
||||||
|
Table 2: Example of Combining Judge and Fan Votes by Rank (Season 1, Week 4)
|
||||||
|
|
||||||
|
<table><tr><td>Contestant</td><td>Total Judges Score</td><td>Judges Score Rank</td><td>Fan Vote*</td><td>Fan Rank*</td><td>Sum of ranks</td></tr><tr><td>Rachel Hunter</td><td>25</td><td>2</td><td>1.1 million</td><td>4</td><td>6</td></tr><tr><td>Joey McIntyre</td><td>20</td><td>4</td><td>3.7 million</td><td>1</td><td>5</td></tr><tr><td>John O’Hurley</td><td>21</td><td>3</td><td>3.2 million</td><td>2</td><td>5</td></tr><tr><td>Kelly Monaco</td><td>26</td><td>1</td><td>2 million</td><td>3</td><td>4</td></tr></table>
|
||||||
|
|
||||||
|
* Fan vote/rank are unknown, hypothetical values chosen to produce the correct final ranks
|
||||||
|
|
||||||
|
# 2. COMBINED BY PERCENT (used for season 3 through $27^{\mathrm{a}}$ )
|
||||||
|
|
||||||
|
Starting in season 3 scores were combined using percents instead of ranks. An example is shown using week 9 of season 5. In that week, Jennie Garth was eliminated. Again, we artificially created fan votes that produce total percents to correctly lead to that result. The judges' percent is computed by dividing the total judge score for the contestant by the sum of total judge scores for all 4 contestants. Based on the judges' percent, Jennie was $3^{\text{rd}}$ . However, adding the percent of the 10 million artificially created fan votes we assigned to the judges' percent she was $4^{\text{th}}$ .
|
||||||
|
|
||||||
|
Table 3: Example of Combining Judge and Fan Votes by Percent (Season 5, Week 9)
|
||||||
|
|
||||||
|
<table><tr><td>Contestant</td><td>Total Judges Score</td><td>Judges Score Percent</td><td>Fan Vote*</td><td>Fan Percent*</td><td>Sum of Percent</td></tr><tr><td>Jennie Garth</td><td>29</td><td>29/117 = 24.8%</td><td>1.1 million</td><td>1.1/10 = 11%</td><td>35.8</td></tr><tr><td>Marie Osmond</td><td>28</td><td>28/117 = 23.9%</td><td>3.7 million</td><td>3.7/10 = 37%</td><td>60.9</td></tr><tr><td>Mel B</td><td>30</td><td>30/117 = 25.6%</td><td>3.2 million</td><td>3.2/10 = 32%</td><td>57.8</td></tr><tr><td>Helio Castroneves</td><td>30</td><td>30/117 = 25.6%</td><td>2 million</td><td>2/10 = 20%</td><td>45.6</td></tr><tr><td>Total</td><td>117</td><td></td><td>10 million</td><td></td><td></td></tr></table>
|
||||||
|
|
||||||
|
* Fan vote is unknown, values hypothetical to produce the correct final standings
|
||||||
|
|
||||||
|
a The year of the return to the rank based method is not known for certain; season 28 is a reasonable assumption.
|
||||||
66
en_D.md
Normal file
66
en_D.md
Normal file
@@ -0,0 +1,66 @@
|
|||||||
|
# 2026 ICM Problem D: Managing Sports for Success
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
"The player's job is to help his team win." - Cliff Blau, baseball historian and statistician "The player's job is to make money for the owner." - all sports team owners
|
||||||
|
|
||||||
|
Fans tend to focus on the players on the field or court, but that is only the tip of the sports business iceberg. Sports are entertainment first and foremost. Entertainment is a profit-generating business and the players are hired mostly for that purpose. Often, fans of spectator sports ignore the financial purpose of a sport and try to focus on the game itself and its participants. However, in professional sports business, the primary goal is to make money for the owner and not necessarily win games. While these two goals may be related, since winning generates more interest in the team, other factors are involved. And for some sports teams, there are crucial moments when opportunity and risk are both high – like this year's situation for teams in the Women's National Basketball Association (WNBA), the most prominent women's professional basketball league in the United States. For many reasons (especially higher fan interest), WNBA teams are hoping to evolve from risky startup businesses into major entertainment enterprises by taking advantage of increased media attention, new team franchises, larger venues, and a new digital platform to increase revenue. The owners in that league need to use sports analytics to succeed on the court but also use financial modeling to achieve significant financial gain in the bottom line of their business's profit sheet.
|
||||||
|
|
||||||
|
Should players (and other employees of the team business) in a sport get paid more for their performance that produces wins or for their contributions in turning a profit for the team owner? Sometimes, a player's sport performance is directly related to profit, but not always. Some players may attract fans based more on popularity than performance. These players may generate ticket, parking, concession, and jersey revenue much more than players with higher levels of performance. Financial and sports analytics models need to connect to create good team decision making.
|
||||||
|
|
||||||
|
In the emerging field of sports analytics with various kinds and amounts of performance data, there continue to be challenges to build statistics that quantify the value of player talents and performances (what statistic to measure, how to measure it, when to measure it). Some players are injured more frequently than others. How does that affect player value? Some have personalities that lead to more popularity and appeal that lead to financial gain. Context and timing matter in the sense that some players, even those with average performance, come through at important moments of the game or critical times in the season. There is a temporal element that must consider the measure of future potential of a player/employee on achieving the goal of the team. Some roles may be performance or skill-based and other roles are accomplished more by hard work and perseverance.
|
||||||
|
|
||||||
|
The player or team perceptions, popularity, timing, and marketing can play major roles, in addition to the location of the team. Teams in large markets often have different sports situations and goals than small-market teams. Those differences impact how owners achieve profit and recruit their players and employees. Can modeling help an owner establish methodologies for setting offers, negotiating, and writing contracts?
|
||||||
|
|
||||||
|
There are many team issues that are strictly or mostly financial, just as there are issues that are mostly sports. In many cases, professional sports teams are franchises that are part of the league enterprise and often operate with additional rules and constraints set by leagues or governments on their player salaries and contracts. These are intended to make the game competition fair with some reasonable amount of competitive balance. Some professional sports have systems that regulate salaries with caps or taxes. Every season, the owner must decide how much to finance with debt versus equity and whether risks in the form of seeking better team performance with associated additional costs are worth taking. In the sports business world, conditions such as revenues, salaries, injuries, trade opportunities, taxes, fees, and interest rates change over time. Sports teams are now seen as premium assets, with values in many sports soaring far beyond historical norms due to financial and market factors such as lucrative media deals and accumulation of vast data streams and intellectual property.
|
||||||
|
|
||||||
|
As a modeling group for a sports team, your ICM team can use publicly available sport and finance data for a team of your choice (the team you select must consist of at least 5 players that play cooperatively at the same time and be a member of a professional league) and build a business and management model for the team for the coming or next season.
|
||||||
|
|
||||||
|
As was mentioned earlier as an example of how this modeling work can be extremely valuable, the WNBA is undergoing significant financial changes -- record viewership, rising franchise values, and significant player benefit expectations. Currently, negotiations and demands over the revenue-sharing agreement between teams and players are sticking points. During this coming season, team owners have an opportunity to remake and improve their business or succumb to risks that may cause them to sell or take on substantial debt. These issues create a situation where solid financial and sport modeling can make a big difference for the current and future owners of these teams. You may use a WNBA team if you care to, but you are not required to do that.
|
||||||
|
|
||||||
|
# Questions to consider:
|
||||||
|
|
||||||
|
Design a dynamic decision-making model that would help your team owner and general managers adjust their leverage in response to changing team performance and economic conditions. The goal is to maximize team profit and value while managing team structure and performance. The model should include priorities and actions for the management teams in both business operations and team operations, and account for systems that will help the owner make decisions through the coming season and beyond.
|
||||||
|
|
||||||
|
Based on the needs of the team and your model, develop a strategy to acquire players for next season using the standard practice for your team's league such as a draft, free agency, trades, transfer fees, or other standard practices. You may want to consider how to value a player or the
|
||||||
|
|
||||||
|
team dynamics in terms of the profit for the team owners. Using the outcomes of your model, discuss the strengths and weaknesses of your strategy on the business.
|
||||||
|
|
||||||
|
There are many league-determined rulings that impact how a single team can operate, such as salary caps, number of players on a roster, schedule (number, order, location, and date of games in a season, so consequently days of rest), media contracts and rights, revenue distributions, and others. If a league is expanding the number of franchises (such as for WNBA), it is likely to impact all teams in the league. Use your model to decide how your team's strategy should change from your initial strategy during a season with league expansion. How does the location for the new team impact your model and resulting strategy? Be clear on the impact on the team owners and locations for the new team that would be particularly harmful or beneficial under an expansion.
|
||||||
|
|
||||||
|
Consider one additional business decision and use your model to design the best strategy for your team. Some examples include but are not limited to:
|
||||||
|
|
||||||
|
- Ticket sales vary greatly by the size of the stadium, time of year, popularity of the team (yours and opponent), size of the team's market, and other factors. A team may choose to maximize ticket sale revenue for each game or lower the prices to have larger attendance with the possibility to convert some of those attendees into season ticket holders. How do you determine the optimal ticket pricing strategy over a season?
|
||||||
|
|
||||||
|
- The venue for the team to play its games may be rented or owned with the need to maintain, renovate, or even build a new venue. How do you balance the long-term cost of the venue when it is a short-term decision?
|
||||||
|
|
||||||
|
- Player equity in ownership can be one strategy for subsidizing large salaries, such as revenue sharing (single season), profit participation (bonus), decision makers (as part of unions or collective bargaining), long-term equity stake (part owner), or other methods. Player equity options need to be sufficiently lucrative for a player to accept it, but not undermine the future funding options. How do you determine which players, if any, are offered equity and how much?
|
||||||
|
|
||||||
|
- Media deals are a large source of revenue, fan engagement, and brand building, often producing high engagement and advertising potential. While leagues usually contract national deals, teams can sometimes broker their own local deals or streaming options. Does your team need to improve or change its media presence?
|
||||||
|
|
||||||
|
- Division or conference structure, which can build or take advantage of rivalries where rival teams play more often, is generally determined by the league. Are there ways that league structures and schedules be reconfigured to increase profit for your team?
|
||||||
|
|
||||||
|
- Determine your own issue that applies to your team or sport and use your model to help decide the issue to improve team performance or owner profit.
|
||||||
|
|
||||||
|
How does your model help management adjust when a key player is injured?
|
||||||
|
|
||||||
|
Write a one- to two-page letter to your team's owner and general manager that summarizes your recommended strategy, discusses trade-offs and risks, and reflects on how your plan supports both competitive success and financial health.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One-to-Two-Page Letter.
|
||||||
|
- References List.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
Competitive balance is how evenly matched the teams are in a league or competition.
|
||||||
|
|
||||||
|
Draft is a way for a sports league to assign new players to teams in an organized manner.
|
||||||
|
|
||||||
|
Free agency is a system that allows players to choose which team they will play for after their contract with a team expires.
|
||||||
78
en_E.md
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78
en_E.md
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|
|||||||
|
# 2026 ICM Problem E: Passive Solar Shading
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
# Background
|
||||||
|
|
||||||
|
Passive solar shading has become a common addition to both housing and commercial buildings as a part of a retrofit or in new construction. It is relatively inexpensive and creates cost savings in heating and cooling. The shades are designed to block summer sun from entering a building, while allowing winter sun to not only enter the building but to warm a thermal mass that can reradiate for many hours after. Strategies such as overhangs, vegetative shading, brise-soleil systems, and high-performance glazing can reduce heat gain in buildings during higher temperatures.
|
||||||
|
|
||||||
|
Passive solar shading is different depending on building orientation, window area distribution between the different faces of the building, and climate. It also requires the presence of an internal thermal mass that can be heated by the direct sun. This thermal mass can be concrete, stone, water, or other material that can store heat. The thermal mass not only stores heat but reduces temperature swings throughout the day.
|
||||||
|
|
||||||
|
These techniques use the predictable path of the sun (determined through the use of solar position calculators), materials, geometry, and natural environmental conditions to maintain comfort and reduce energy consumption. However, the typical calculations make use of the angle of the sun at solar noon on the Summer and Winter Solstices to calculate the optimal extension of a shade over a window as shown in Figure 1. This is a simplistic view of the problem, and future metrics must do better to account for change.
|
||||||
|
|
||||||
|
|
||||||
|
Figure 1: Passive Solar Shading - Winter and Summer Sun on Solstices
|
||||||
|
|
||||||
|
# Scenario
|
||||||
|
|
||||||
|
You have been hired by the Collective Organizations Making Astrophysical Protections (COMAP) to innovate the next generation of solar shading strategies to be implemented at the notional Sungrove University and notional Borealis University.
|
||||||
|
|
||||||
|
The notional Sungrove University, located in a warm, low-latitude region with high solar exposure and increasingly frequent heat waves, is planning a major transformation of its main academic quad. The campus currently suffers from excessive cooling costs and glare in the classrooms. The university leadership has decided to pursue a net-zero cooling initiative by 2040.
|
||||||
|
|
||||||
|
Notably, Sungrove University is planning to retrofit its Academic Hall North. It is a two-story classroom and office building. The interior layout combines perimeter offices and classrooms with interior corridors. The building has a rectangular footprint (60m × 24m) with its long side aligned east-west as shown in Figure 2. The facade consists of double glazing and a brick veneer with an average window-to-wall ratio of $45\%$ on the south facing side and $30\%$ on the remaining sides. The building relies on mechanical cooling in the summer and hydronic heating in the winter with limited passive strategies in place. Additional features of this notional building are yours to imagine. Ensure you communicate these features in your writing to COMAP.
|
||||||
|
|
||||||
|
|
||||||
|
Figure 2: Academic Hall North footprint
|
||||||
|
|
||||||
|
Additionally, COMAP has been hired by the notional Borealis University, located at a high latitude where winter temperatures remain below freezing for months, sunlight hours are limited, and buildings experience heavy heating demands.
|
||||||
|
|
||||||
|
Sungrove University and Borealis University are also both planning a new student union that will serve as the hub of university activities. They have each mandated that their new student union building relies heavily on passive solar shading rather than mechanical cooling systems. The Universities want their student union building to serve as a prototype for future developments, meaning that their passive solar strategy design must perform well not only today, but under projected climate conditions well into the future.
|
||||||
|
|
||||||
|
Beyond the standard approach to shading as outlined in the Background, to assist these notional universities, you should extend your ideas to include:
|
||||||
|
|
||||||
|
- Shading needs throughout the day rather than just at solar noon.
|
||||||
|
- Windows of different sizes and shapes.
|
||||||
|
- Windows that do not face exactly south/north (depending on the hemisphere).
|
||||||
|
- Shades of different styles and materials that would match the architecture of the building.
|
||||||
|
|
||||||
|
As with any new strategy or model, you will not only need to describe your approach but also explain the advantages that your proposal holds over the previous standard. COMAP needs to know how your passive solar shading strategies can more effectively reduce heat gain in campus buildings during the summer while still admitting beneficial winter sun.
|
||||||
|
|
||||||
|
# Requirements
|
||||||
|
|
||||||
|
Your team has been asked by COMAP to provide a model-based feasibility analysis that determines how Sungrove University can reduce its academic year cooling load with passive solar design in the retrofit of buildings on campus. To do so, design a retrofit for Sungrove University's Academic Hall North that optimizes heating and cooling throughout the academic year. What passive solar strategies and building features would you use, and how would you evaluate their performance?
|
||||||
|
|
||||||
|
Borealis University has a building with a similar design to Sungrove University's Academic Hall North. How can extending your work for Sungrove University to include the crucial importance of the effective use of a thermal mass provide Borealis University with a plan to use passive solar shading? You may want to consider building geometry, material selection, glazing positioning, internal thermal mass, or other aspects to maximize winter heat gain while avoiding overheating in the warmer months.
|
||||||
|
|
||||||
|
The retrofit design models at both Sungrove and Borealis Universities are helpful for only those notional sites. Adapt your model and discuss the design considerations for other locations including the different heating and cooling needs at places that might have similar latitudes.
|
||||||
|
|
||||||
|
Design a passive solar shading strategy for the new student union building at either Sungrove University or Borealis University that keeps the building temperate. Describe the strategies, building features, and modeling approaches you would use to evaluate performance over time. You may wish to address some of the following in your analysis:
|
||||||
|
|
||||||
|
- Predicting solar heat gain
|
||||||
|
- Estimating heating and/or cooling load reductions
|
||||||
|
- Accounting for seasonal variations
|
||||||
|
- Evaluating the tradeoffs between daylighting needs and shading effectiveness
|
||||||
|
|
||||||
|
Write a one-to-two-page letter to either Sungrove University or Borealis University (not both) outlining the steps they should take to include passive solar shading in both their retrofit and new building plans.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
One-to-Two-Page Letter.
|
||||||
|
- References List.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
|
|
||||||
|
# Glossary
|
||||||
|
|
||||||
|
Solar noon is the moment during the day when the Sun is at its highest point in the sky for a given location.
|
||||||
|
|
||||||
|
Winter Solstice is the day with the least daylight of the year, caused by Earth's tilt.
|
||||||
|
|
||||||
|
Summer Solstice is the day with the most daylight of the year, caused by Earth's tilt.
|
||||||
|
|
||||||
|
Notional means theoretical or fictitious. The universities in this problem are not real, but only theoretical case studies.
|
||||||
|
|
||||||
|
Net-zero cooling means providing cooling without adding greenhouse gases to the atmosphere.
|
||||||
39
en_F.md
Normal file
39
en_F.md
Normal file
@@ -0,0 +1,39 @@
|
|||||||
|
# 2026 ICM
|
||||||
|
|
||||||
|
# Problem F: To Gen-AI, or Not To Gen-AI (or how to Gen-AI)? That is the Question!
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
In just a few years, Generative Artificial Intelligence (Gen-AI) has gone from a tool of limited capacity, only used by a few early-adopters to a powerful and inescapable resource embedded in our daily lives. Over time, research suggests that Gen-AI could impact the future of work. For example, in some fields, Gen-AI may replace humans (or heavily reduce the human workload), while other fields might not be heavily impacted or might even grow.
|
||||||
|
|
||||||
|
In this question, you will explore how post-secondary educational institutions, of various types, should best prepare their future graduates in light of this new technology. Specifically, you are asking do to the following.
|
||||||
|
|
||||||
|
- Choose three careers, one from each of the following categories:
|
||||||
|
|
||||||
|
STEM career: people in this career often have at least a four year university degree in the sciences, engineering, or mathematics;
|
||||||
|
Trade career: people in this career often have training from a trade school and/or an apprenticeship program, such as chef, plumber, and electrician;
|
||||||
|
○ Arts career: people in this career often have studied at an arts school, conservatory, or cultural center, such as musician, dancer, or painter.
|
||||||
|
|
||||||
|
- Design a data-informed model to explore the future of each of your three chosen professions, given the current trajectory and expected impacts of Gen-AI. Be sure to identify your data sources as well as your reasoning behind any drivers that you expect to change this profession as a result of Gen-AI. Note: you may leverage existing research on the future of work, but be sure to cite your sources and explain how you are using the established research to inform your analysis.
|
||||||
|
|
||||||
|
- Identify a specific post-secondary institution and program of study for each career you are analyzing (one at a university, one at a trade school, one at an arts school), and focus your recommendations accordingly. In other words, you should have three sets of recommendations that address the following question: Based on your analysis, how would you advise the leaders of each of these institutions to address Gen-AI in the programs specific to the careers you are analyzing?
|
||||||
|
|
||||||
|
Below are just some thoughts you may wish to consider; teams should not attempt to address all of these ideas, but should use these as inspiration that will lead to a cogent and thorough analysis that should vary team to team.
|
||||||
|
|
||||||
|
○ Should the program of study grow or shrink (graduate more or fewer people) as a result of changes in the career due to Gen-AI? If the field should grow, how might the institution recruit more people; and if the field should shrink, are there other programs in the school that should grow to absorb the people who used to study in this program?
|
||||||
|
|
||||||
|
○ What should these three different programs of study teach about Gen-AI? Many post-secondary institutions of learning have asked this question and are still developing their response. While some institutions have outright banned the use of AI on any assignments, others have brought the use of AI to the forefront of their curriculum. Some schools aim to produce experts who can contribute to the leading edge of the technological field, while some focus on graduating students in non-technical fields who are fluent users of the technology. Some institutions encourage their students to think about all the ways they can apply this new technology, and some schools challenge students to carefully weigh the benefits and costs of using AI, given the requisite energy usage, water demands, and risk of insufficient (often missing or incorrect) attribution to the original creators of ideas or content. For the three programs of study at the three institutions you've selected, what do you recommend to best support the employability of their graduates? Be sure to support your recommendations with the results of a mathematical model.
|
||||||
|
|
||||||
|
While this problem poses the question through the context of employability of graduates in a world where Gen-AI is ubiquitous, perhaps employment demands are not the only way to measure the success of the institutional policies you are proposing. What other factors do you believe should be considered, and how do your models and recommendations change when you consider these other factors?
|
||||||
|
|
||||||
|
If you believe your specific recommendations can be generalized beyond one institution and/or beyond one program, be sure to explain the extent of the generalization and justify this.
|
||||||
|
|
||||||
|
Your PDF solution of no more than 25 total pages should include:
|
||||||
|
|
||||||
|
One-page Summary Sheet.
|
||||||
|
Table of Contents.
|
||||||
|
- Your complete solution.
|
||||||
|
- References list.
|
||||||
|
AI Use Report (If used does not count toward the 25-page limit.)
|
||||||
|
|
||||||
|
Note: There is no specific required minimum page length for a complete ICM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.
|
||||||
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Reference in New Issue
Block a user