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  • Hb#cu+cL4S zyurKwdOc=3y1&JMje&vX_dc^Ru>4W;?Js?KGx3%b|1Jak?k_$4TY~tT-8UI4>u>#| zXJ%w$`^^SDGk}5a_qNPz^uLceJu@34)9=?~VPt0eEiUvdjI3{&`(OI}hm7fOG5{UJ zZ*$|#4LY{J>oNX5=5Km{Kh|pgJyvh-GW;zDEUZi{zs-X;Jr<_FwPmBDd&}GZay>Q{ zroY9H?JcIi_nD2A{*NM#e{IW3|NHpTv$4KS{J-e|{uWa#%OOVAb;nv)Y2npPeNd@E!4$HLXy$vnuJ71 JMi>_Qe*ql55kCL` literal 0 HcmV?d00001 diff --git a/2026_ICM_Problem_D.pdf-5741bf85-1250-4e9c-bd07-1ca6a4d444b1/full.md b/2026_ICM_Problem_D.pdf-5741bf85-1250-4e9c-bd07-1ca6a4d444b1/full.md new file mode 100644 index 0000000..9ac091f --- /dev/null +++ b/2026_ICM_Problem_D.pdf-5741bf85-1250-4e9c-bd07-1ca6a4d444b1/full.md @@ -0,0 +1,66 @@ +# 2026 ICM Problem D: Managing Sports for Success + +![](images/0af13f43c25137d8e22b7c835aa3d8dc127f8498eea06196deb71b9e795dbf0c.jpg) + +"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. \ No newline at end of file diff --git a/2026_ICM_Problem_D.pdf-5741bf85-1250-4e9c-bd07-1ca6a4d444b1/images/0af13f43c25137d8e22b7c835aa3d8dc127f8498eea06196deb71b9e795dbf0c.jpg b/2026_ICM_Problem_D.pdf-5741bf85-1250-4e9c-bd07-1ca6a4d444b1/images/0af13f43c25137d8e22b7c835aa3d8dc127f8498eea06196deb71b9e795dbf0c.jpg new file mode 100644 index 0000000000000000000000000000000000000000..ec4946c61884764bda79bd703d8488ee0fbd6c37 GIT binary patch literal 33536 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E: Passive Solar Shading + +![](images/0853598daca459a320840f7e63c055ef2d8c80fa5d1e705244e5d9e1d7fe417c.jpg) + +# 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. + +![](images/3c7807f15520a08ceecdd4931bf673e0dae548121e7ad69484567853119941f4.jpg) +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. + +![](images/866cfa696b4ce3f50d373d3bcab45c4c2feee7d2fb16dfb2490d21e0f75e6b03.jpg) +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. 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+ ] + } + ], + "index": 0 + }, + { + "type": "image", + "bbox": [ + 173, + 112, + 436, + 228 + ], + "blocks": [ + { + "bbox": [ + 173, + 112, + 436, + 228 + ], + "lines": [ + { + "bbox": [ + 173, + 112, + 436, + 228 + ], + "spans": [ + { + "bbox": [ + 173, + 112, + 436, + 228 + ], + "type": "image", + "image_path": "0853598daca459a320840f7e63c055ef2d8c80fa5d1e705244e5d9e1d7fe417c.jpg" + } + ] + } + ], + "index": 1, + "angle": 0, + "type": "image_body" + } + ], + "index": 1 + }, + { + "bbox": [ + 71, + 243, + 136, + 257 + ], + "type": "title", + "angle": 0, + "lines": [ + { + "bbox": [ + 71, + 243, + 136, + 257 + ], + "spans": [ + { + "bbox": [ + 71, + 243, + 136, + 257 + ], + "type": "text", + "content": "Background" + } + ] + } + ], + "index": 2 + }, + { + "bbox": [ + 70, + 264, + 536, + 361 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 70, + 264, + 536, + 361 + ], + "spans": [ + { + "bbox": [ + 70, + 264, + 536, + 361 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 3 + }, + { + "bbox": [ + 70, + 369, + 523, + 438 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 70, + 369, + 523, + 438 + ], + "spans": [ + { + "bbox": [ + 70, + 369, + 523, + 438 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 4 + }, + { + "bbox": [ + 70, + 445, + 541, + 529 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 70, + 445, + 541, + 529 + ], + "spans": [ + { + "bbox": [ + 70, + 445, + 541, + 529 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 5 + }, + { + "type": "image", + "bbox": [ + 217, + 536, + 394, + 705 + ], + "blocks": [ + { + "bbox": [ + 217, + 536, + 394, + 705 + ], + "lines": [ + { + "bbox": [ + 217, + 536, + 394, + 705 + ], + "spans": [ + { + "bbox": [ + 217, + 536, + 394, + 705 + ], + "type": "image", + "image_path": "3c7807f15520a08ceecdd4931bf673e0dae548121e7ad69484567853119941f4.jpg" + } + ] + } + ], + "index": 6, + "angle": 0, + "type": "image_body" + }, + { + "bbox": [ + 159, + 707, + 452, + 719 + ], + "lines": [ + { + "bbox": [ + 159, + 707, + 452, + 719 + ], + "spans": [ + { + "bbox": [ + 159, + 707, + 452, + 719 + ], + "type": "text", + "content": "Figure 1: Passive Solar Shading - Winter and Summer Sun on Solstices" + } + ] + } + ], + "index": 7, + "angle": 0, + "type": "image_caption" + } + ], + "index": 6 + } + ], + "discarded_blocks": [ + { + "bbox": [ + 140, + 744, + 470, + 757 + ], + "type": "footer", + "angle": 0, + "lines": [ + { + "bbox": [ + 140, + 744, + 470, + 757 + ], + "spans": [ + { + "bbox": [ + 140, + 744, + 470, + 757 + ], + "type": "text", + "content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org" + } + ] + } + ], + "index": 8 + } + ], + "page_size": [ + 612, + 792 + ], + "page_idx": 0 + }, + { + "para_blocks": [ + { + "bbox": [ + 71, + 72, + 116, + 85 + ], + "type": "title", + "angle": 0, + "lines": [ + { + "bbox": [ + 71, + 72, + 116, + 85 + ], + "spans": [ + { + "bbox": [ + 71, + 72, + 116, + 85 + ], + "type": "text", + "content": "Scenario" + } + ] + } + ], + "index": 0 + }, + { + "bbox": [ + 71, + 94, + 523, + 135 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 71, + 94, + 523, + 135 + ], + "spans": [ + { + "bbox": [ + 71, + 94, + 523, + 135 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 1 + }, + { + "bbox": [ + 71, + 143, + 522, + 211 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 71, + 143, + 522, + 211 + ], + "spans": [ + { + "bbox": [ + 71, + 143, + 522, + 211 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 2 + }, + { + "bbox": [ + 70, + 220, + 539, + 331 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 70, + 220, + 539, + 331 + ], + "spans": [ + { + "bbox": [ + 70, + 220, + 539, + 331 + ], + "type": "text", + "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 " + }, + { + "bbox": [ + 70, + 220, + 539, + 331 + ], + "type": "inline_equation", + "content": "45\\%" + }, + { + "bbox": [ + 70, + 220, + 539, + 331 + ], + "type": "text", + "content": " on the south facing side and " + }, + { + "bbox": [ + 70, + 220, + 539, + 331 + ], + "type": "inline_equation", + "content": "30\\%" + }, + { + "bbox": [ + 70, + 220, + 539, + 331 + ], + "type": "text", + "content": " 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." + } + ] + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 166, + 347, + 440, + 453 + ], + "blocks": [ + { + "bbox": [ + 166, + 347, + 440, + 453 + ], + "lines": [ + { + "bbox": [ + 166, + 347, + 440, + 453 + ], + "spans": [ + { + "bbox": [ + 166, + 347, + 440, + 453 + ], + "type": "image", + "image_path": "866cfa696b4ce3f50d373d3bcab45c4c2feee7d2fb16dfb2490d21e0f75e6b03.jpg" + } + ] + } + ], + "index": 4, + "angle": 0, + "type": "image_body" + }, + { + "bbox": [ + 222, + 456, + 389, + 469 + ], + "lines": [ + { + "bbox": [ + 222, + 456, + 389, + 469 + ], + "spans": [ + { + "bbox": [ + 222, + 456, + 389, + 469 + ], + "type": "text", + "content": "Figure 2: Academic Hall North footprint" + } + ] + } + ], + "index": 5, + "angle": 0, + "type": "image_caption" + } + ], + "index": 4 + }, + { + "bbox": [ + 71, + 476, + 538, + 518 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 71, + 476, + 538, + 518 + ], + "spans": [ + { + "bbox": [ + 71, + 476, + 538, + 518 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 6 + }, + { + "bbox": [ + 70, + 525, + 539, + 609 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 70, + 525, + 539, + 609 + ], + "spans": [ + { + "bbox": [ + 70, + 525, + 539, + 609 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 7 + }, + { + "bbox": [ + 71, + 616, + 532, + 644 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 71, + 616, + 532, + 644 + ], + "spans": [ + { + "bbox": [ + 71, + 616, + 532, + 644 + ], + "type": "text", + "content": "Beyond the standard approach to shading as outlined in the Background, to assist these notional universities, you should extend your ideas to include:" + } + ] + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 653, + 535, + 711 + ], + "type": "list", + "angle": 0, + "index": 13, + "blocks": [ + { + "bbox": [ + 86, + 653, + 414, + 666 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 653, + 414, + 666 + ], + "spans": [ + { + "bbox": [ + 86, + 653, + 414, + 666 + ], + "type": "text", + "content": "- Shading needs throughout the day rather than just at solar noon." + } + ] + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 668, + 295, + 681 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 668, + 295, + 681 + ], + "spans": [ + { + "bbox": [ + 86, + 668, + 295, + 681 + ], + "type": "text", + "content": "- Windows of different sizes and shapes." + } + ] + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 682, + 481, + 696 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 682, + 481, + 696 + ], + "spans": [ + { + "bbox": [ + 86, + 682, + 481, + 696 + ], + "type": "text", + "content": "- Windows that do not face exactly south/north (depending on the hemisphere)." + } + ] + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 697, + 535, + 711 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 697, + 535, + 711 + ], + "spans": [ + { + "bbox": [ + 86, + 697, + 535, + 711 + ], + "type": "text", + "content": "- Shades of different styles and materials that would match the architecture of the building." + } + ] + } + ], + "index": 12 + } + ], + "sub_type": "text" + } + ], + "discarded_blocks": [ + { + "bbox": [ + 141, + 744, + 470, + 757 + ], + "type": "footer", + "angle": 0, + "lines": [ + { + "bbox": [ + 141, + 744, + 470, + 757 + ], + "spans": [ + { + "bbox": [ + 141, + 744, + 470, + 757 + ], + "type": "text", + "content": "©2026 by COMAP | www.mathmodels.org | info@comap.org |" + } + ] + } + ], + "index": 14 + } + ], + "page_size": [ + 612, + 792 + ], + "page_idx": 1 + }, + { + "para_blocks": [ + { + "bbox": [ + 67, + 72, + 535, + 128 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 72, + 535, + 128 + ], + "spans": [ + { + "bbox": [ + 67, + 72, + 535, + 128 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 0 + }, + { + "bbox": [ + 69, + 136, + 145, + 149 + ], + "type": "title", + "angle": 0, + "lines": [ + { + "bbox": [ + 69, + 136, + 145, + 149 + ], + "spans": [ + { + "bbox": [ + 69, + 136, + 145, + 149 + ], + "type": "text", + "content": "Requirements" + } + ] + } + ], + "index": 1 + }, + { + "bbox": [ + 67, + 156, + 526, + 240 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 156, + 526, + 240 + ], + "spans": [ + { + "bbox": [ + 67, + 156, + 526, + 240 + ], + "type": "text", + "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?" + } + ] + } + ], + "index": 2 + }, + { + "bbox": [ + 67, + 253, + 541, + 336 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 253, + 541, + 336 + ], + "spans": [ + { + "bbox": [ + 67, + 253, + 541, + 336 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 3 + }, + { + "bbox": [ + 67, + 344, + 534, + 387 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 344, + 534, + 387 + ], + "spans": [ + { + "bbox": [ + 67, + 344, + 534, + 387 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 4 + }, + { + "bbox": [ + 67, + 394, + 526, + 456 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 394, + 526, + 456 + ], + "spans": [ + { + "bbox": [ + 67, + 394, + 526, + 456 + ], + "type": "text", + "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. 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The universities in this problem are not real, but only theoretical case studies." + } + ] + } + ], + "index": 5 + }, + { + "bbox": [ + 68, + 331, + 530, + 346 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 68, + 331, + 530, + 346 + ], + "spans": [ + { + "bbox": [ + 68, + 331, + 530, + 346 + ], + "type": "text", + "content": "Net-zero cooling means providing cooling without adding greenhouse gases to the atmosphere." + } + ] + } + ], + "index": 6 + } + ], + "discarded_blocks": [ + { + "bbox": [ + 138, + 743, + 470, + 757 + ], + "type": "footer", + "angle": 0, + "lines": [ + { + "bbox": [ + 138, + 743, + 470, + 757 + ], + "spans": [ + { + "bbox": [ + 138, + 743, + 470, + 757 + ], + "type": "text", + "content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org" + } + ] + } + ], + "index": 7 + } + ], + "page_size": [ + 612, + 792 + ], + "page_idx": 3 + } + ], + "_backend": "vlm", + "_version_name": "2.6.4" +} \ No newline at end of file diff --git a/2026_ICM_Problem_F.pdf-f76a8376-8020-4dc3-9958-3024f9e017f9/876efaab-1368-4907-adb9-f87cbf197a44_content_list.json b/2026_ICM_Problem_F.pdf-f76a8376-8020-4dc3-9958-3024f9e017f9/876efaab-1368-4907-adb9-f87cbf197a44_content_list.json new file mode 100644 index 0000000..73558ac --- /dev/null +++ b/2026_ICM_Problem_F.pdf-f76a8376-8020-4dc3-9958-3024f9e017f9/876efaab-1368-4907-adb9-f87cbf197a44_content_list.json @@ -0,0 +1,227 @@ +[ + { + "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", + "text": "©2026 by COMAP | www.mathmodels.org | www.mathmodels.org | info@comap.org |", + "bbox": [ + 223, + 938, + 769, + 955 + ], + "page_idx": 1 + } +] \ No newline at end of file diff --git a/2026_ICM_Problem_F.pdf-f76a8376-8020-4dc3-9958-3024f9e017f9/876efaab-1368-4907-adb9-f87cbf197a44_model.json b/2026_ICM_Problem_F.pdf-f76a8376-8020-4dc3-9958-3024f9e017f9/876efaab-1368-4907-adb9-f87cbf197a44_model.json new file mode 100644 index 0000000..7dab097 --- /dev/null +++ b/2026_ICM_Problem_F.pdf-f76a8376-8020-4dc3-9958-3024f9e017f9/876efaab-1368-4907-adb9-f87cbf197a44_model.json @@ -0,0 +1,303 @@ +[ + [ + { + "type": "title", + "bbox": [ + 0.456, + 0.092, + 0.543, + 0.107 + ], + "angle": 0, + "content": "2026 ICM" + }, + { + "type": "title", + "bbox": [ + 0.144, + 0.11, + 0.855, + 0.126 + ], + "angle": 0, + "content": "Problem F: To Gen-AI, or Not To Gen-AI (or how to Gen-AI)? That is the Question!" + }, + { + "type": "image", + "bbox": [ + 0.328, + 0.141, + 0.67, + 0.32 + ], + "angle": 0, + "content": null + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.347, + 0.875, + 0.435 + ], + "angle": 0, + "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." + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.445, + 0.869, + 0.498 + ], + "angle": 0, + "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." + }, + { + "type": "text", + "bbox": [ + 0.143, + 0.507, + 0.684, + 0.522 + ], + "angle": 0, + "content": "- Choose three careers, one from each of the following categories:" + }, + { + "type": "text", + "bbox": [ + 0.201, + 0.525, + 0.829, + 0.557 + ], + "angle": 0, + "content": "STEM career: people in this career often have at least a four year university degree in the sciences, engineering, or mathematics;" + }, + { + "type": "text", + "bbox": [ + 0.201, + 0.56, + 0.867, + 0.592 + ], + "angle": 0, + "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;" + }, + { + "type": "text", + "bbox": [ + 0.201, + 0.595, + 0.775, + 0.626 + ], + "angle": 0, + "content": "○ Arts career: people in this career often have studied at an arts school, conservatory, or cultural center, such as musician, dancer, or painter." + }, + { + "type": "list", + "bbox": [ + 0.201, + 0.525, + 0.867, + 0.626 + ], + "angle": 0, + "content": null + }, + { + "type": "text", + "bbox": [ + 0.143, + 0.63, + 0.882, + 0.733 + ], + "angle": 0, + "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." + }, + { + "type": "text", + "bbox": [ + 0.143, + 0.734, + 0.872, + 0.837 + ], + "angle": 0, + "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?" + }, + { + "type": "text", + "bbox": [ + 0.17, + 0.839, + 0.882, + 0.89 + ], + "angle": 0, + "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." + }, + { + "type": "footer", + "bbox": [ + 0.225, + 0.939, + 0.769, + 0.957 + ], + "angle": 0, + "content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org" + } + ], + [ + { + "type": "text", + "bbox": [ + 0.201, + 0.092, + 0.88, + 0.177 + ], + "angle": 0, + "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?" + }, + { + "type": "text", + "bbox": [ + 0.202, + 0.179, + 0.868, + 0.436 + ], + "angle": 0, + "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." + }, + { + "type": "text", + "bbox": [ + 0.201, + 0.44, + 0.88, + 0.526 + ], + "angle": 0, + "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?" + }, + { + "type": "text", + "bbox": [ + 0.202, + 0.528, + 0.842, + 0.579 + ], + "angle": 0, + "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." + }, + { + "type": "text", + "bbox": [ + 0.113, + 0.616, + 0.595, + 0.634 + ], + "angle": 0, + "content": "Your PDF solution of no more than 25 total pages should include:" + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.652, + 0.386, + 0.667 + ], + "angle": 0, + "content": "One-page Summary Sheet." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.673, + 0.321, + 0.687 + ], + "angle": 0, + "content": "Table of Contents." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.694, + 0.364, + 0.709 + ], + "angle": 0, + "content": "- Your complete solution." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.715, + 0.297, + 0.729 + ], + "angle": 0, + "content": "- References list." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.736, + 0.685, + 0.752 + ], + "angle": 0, + "content": "AI Use Report (If used does not count toward the 25-page limit.)" + }, + { + "type": "list", + "bbox": [ + 0.142, + 0.652, + 0.685, + 0.752 + ], + "angle": 0, + "content": null + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.77, + 0.882, + 0.899 + ], + "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." + }, + { + "type": "footer", + "bbox": [ + 0.225, + 0.939, + 0.771, + 0.957 + ], + "angle": 0, + "content": "©2026 by COMAP | www.mathmodels.org | www.mathmodels.org | info@comap.org |" + } + ] +] \ No newline at end of file diff --git a/2026_ICM_Problem_F.pdf-f76a8376-8020-4dc3-9958-3024f9e017f9/876efaab-1368-4907-adb9-f87cbf197a44_origin.pdf b/2026_ICM_Problem_F.pdf-f76a8376-8020-4dc3-9958-3024f9e017f9/876efaab-1368-4907-adb9-f87cbf197a44_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..34bbac59a88871c8ef2d1f3a31a9ec7c2808146e GIT binary patch literal 203534 zcma%hV|bn2wsz1ajh%*#)nLW8ZQJ&W+1R#i+iDuCabw$d@}+M#-hF-hT<5Gm&w9q3 zW6m+gyr;(fkjM%OQ`6C~z(JGb)b-87LDS*W;@jw(!$EU$0%Y|}jDYy`?-F@@fTF9d z5kNrCLC@00gqs@<+Q{1QuX?|W|EVBg<7n-GPY;kVGX!ejGybKB&%pB6Q|nI)f3^8{ 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That is the Question! + +![](images/e22c5822463cf6f31feee50f4cc9573edbf6d99fa928c613ab5390964ee32cd6.jpg) + +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. 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b/2026_MCM-ICM_Problems/MCM-ICM_Summary.tex new file mode 100644 index 0000000..7067302 --- /dev/null +++ b/2026_MCM-ICM_Problems/MCM-ICM_Summary.tex @@ -0,0 +1,76 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% 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 diff --git a/2026_MCM_Problem_A.pdf-fe7a11fc-0258-4218-b8a9-efe5673983f8/5ac4e770-8c9f-48d1-949e-b97744a7eaeb_content_list.json b/2026_MCM_Problem_A.pdf-fe7a11fc-0258-4218-b8a9-efe5673983f8/5ac4e770-8c9f-48d1-949e-b97744a7eaeb_content_list.json new file mode 100644 index 0000000..dbde98d --- /dev/null +++ b/2026_MCM_Problem_A.pdf-fe7a11fc-0258-4218-b8a9-efe5673983f8/5ac4e770-8c9f-48d1-949e-b97744a7eaeb_content_list.json @@ -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, + 955 + ], + "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, + 90, + 854, + 167 + ], + "page_idx": 1 + }, + { + "type": "list", + "sub_type": "text", + "list_items": [ + "○ 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?" + ], + "bbox": [ + 200, + 170, + 846, + 250 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3. Sensitivity and Assumptions: Examine how your predictions vary after making changes in your modeling assumptions, parameter values, and fluctuations in usage patterns.", + "bbox": [ + 140, + 271, + 879, + 310 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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.", + "bbox": [ + 140, + 330, + 874, + 470 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Your report should present:", + "text_level": 1, + "bbox": [ + 112, + 491, + 354, + 508 + ], + "page_idx": 1 + }, + { + "type": "list", + "sub_type": "text", + "list_items": [ + "- 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." + ], + "bbox": [ + 171, + 518, + 741, + 635 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "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.", + "bbox": [ + 109, + 659, + 870, + 758 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Your PDF solution of no more than 25 total pages should include:", + "bbox": [ + 112, + 784, + 635, + 801 + ], + "page_idx": 1 + }, + { + "type": "list", + "sub_type": "text", + "list_items": [ + "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.)" + ], + "bbox": [ + 140, + 804, + 684, + 895 + ], + "page_idx": 1 + }, + { + "type": "footer", + "text": "©2026 by COMAP | www.mathmodels.org | info@comap.org |", + "bbox": [ + 223, + 938, + 769, + 955 + ], + "page_idx": 1 + }, + { + "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": [ + 109, + 90, + 880, + 213 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Glossary", + "text_level": 1, + "bbox": [ + 112, + 233, + 192, + 251 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Smartphone: is a mobile device that combines the functionality of a traditional cell phone with advanced computing capabilities.", + "bbox": [ + 109, + 272, + 870, + 310 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Power Consumption: the rate at which a device uses electrical energy from its battery or power source.", + "bbox": [ + 109, + 333, + 874, + 369 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Processor Load: the actual amount of work being done by the processor at a given moment.", + "bbox": [ + 112, + 393, + 843, + 411 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "State of Charge (SOC): a measure of how much energy remains in a battery compared to its full capacity, expressed as a percentage.", + "bbox": [ + 109, + 433, + 880, + 472 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Time-to-Empty: the estimated amount of time remaining before a battery is completely discharged.", + "bbox": [ + 109, + 493, + 810, + 532 + ], + "page_idx": 2 + }, + { + "type": "footer", + "text": "©2026 by COMAP | www.mathmodels.org | info@comap.org |", + "bbox": [ + 225, + 938, + 769, + 955 + ], + "page_idx": 2 + } +] \ No newline at end of file diff --git a/2026_MCM_Problem_A.pdf-fe7a11fc-0258-4218-b8a9-efe5673983f8/5ac4e770-8c9f-48d1-949e-b97744a7eaeb_model.json b/2026_MCM_Problem_A.pdf-fe7a11fc-0258-4218-b8a9-efe5673983f8/5ac4e770-8c9f-48d1-949e-b97744a7eaeb_model.json new file mode 100644 index 0000000..42e73c9 --- /dev/null +++ b/2026_MCM_Problem_A.pdf-fe7a11fc-0258-4218-b8a9-efe5673983f8/5ac4e770-8c9f-48d1-949e-b97744a7eaeb_model.json @@ -0,0 +1,514 @@ +[ + [ + { + "type": "title", + "bbox": [ + 0.449, + 0.091, + 0.549, + 0.108 + ], + "angle": 0, + "content": "2026 MCM" + }, + { + "type": "title", + "bbox": [ + 0.292, + 0.111, + 0.707, + 0.129 + ], + "angle": 0, + "content": "Problem A: Modeling Smartphone Battery Drain" + }, + { + "type": "image", + "bbox": [ + 0.338, + 0.164, + 0.408, + 0.262 + ], + "angle": 0, + "content": null + }, + { + "type": "image_caption", + "bbox": [ + 0.354, + 0.263, + 0.392, + 0.276 + ], + "angle": 0, + "content": "25%" + }, + { + "type": "image", + "bbox": [ + 0.421, + 0.164, + 0.49, + 0.262 + ], + "angle": 0, + "content": null + }, + { + "type": "image_caption", + "bbox": [ + 0.436, + 0.264, + 0.475, + 0.276 + ], + "angle": 0, + "content": "50%" + }, + { + "type": "image", + "bbox": [ + 0.502, + 0.164, + 0.573, + 0.262 + ], + "angle": 0, + "content": null + }, + { + "type": "image_caption", + "bbox": [ + 0.52, + 0.264, + 0.558, + 0.276 + ], + "angle": 0, + "content": "75%" + }, + { + "type": "image", + "bbox": [ + 0.585, + 0.164, + 0.657, + 0.262 + ], + "angle": 0, + "content": null + }, + { + "type": "image_caption", + "bbox": [ + 0.597, + 0.264, + 0.644, + 0.276 + ], + "angle": 0, + "content": "100%" + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.322, + 0.881, + 0.502 + ], + "angle": 0, + "content": "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." + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.51, + 0.861, + 0.59 + ], + "angle": 0, + "content": "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." + }, + { + "type": "title", + "bbox": [ + 0.114, + 0.618, + 0.241, + 0.636 + ], + "angle": 0, + "content": "Requirements:" + }, + { + "type": "text", + "bbox": [ + 0.143, + 0.645, + 0.867, + 0.744 + ], + "angle": 0, + "content": "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." + }, + { + "type": "text", + "bbox": [ + 0.17, + 0.766, + 0.879, + 0.905 + ], + "angle": 0, + "content": "Data as support, not substitute: You may collect or use data for parameter estimation and validation. 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Compare predictions to observed or plausible behavior, quantify uncertainty, and identify where the model performs well or poorly." + }, + { + "type": "text", + "bbox": [ + 0.201, + 0.171, + 0.848, + 0.209 + ], + "angle": 0, + "content": "○ Show how your model explains differences in these outcomes and identify the specific drivers of rapid battery drain in each case." + }, + { + "type": "text", + "bbox": [ + 0.201, + 0.211, + 0.84, + 0.251 + ], + "angle": 0, + "content": "- Which activities or conditions produce the greatest reductions in battery life? Which ones change the model surprisingly little?" + }, + { + "type": "list", + "bbox": [ + 0.201, + 0.171, + 0.848, + 0.251 + ], + "angle": 0, + "content": null + }, + { + "type": "text", + "bbox": [ + 0.141, + 0.272, + 0.88, + 0.311 + ], + "angle": 0, + "content": "3. Sensitivity and Assumptions: Examine how your predictions vary after making changes in your modeling assumptions, parameter values, and fluctuations in usage patterns." + }, + { + "type": "text", + "bbox": [ + 0.141, + 0.332, + 0.875, + 0.471 + ], + "angle": 0, + "content": "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? 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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." + }, + { + "type": "text", + "bbox": [ + 0.113, + 0.785, + 0.637, + 0.803 + ], + "angle": 0, + "content": "Your PDF solution of no more than 25 total pages should include:" + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.805, + 0.387, + 0.821 + ], + "angle": 0, + "content": "One-page Summary Sheet." + }, + { + "type": "text", + "bbox": [ + 0.143, + 0.823, + 0.323, + 0.838 + ], + "angle": 0, + "content": "Table of Contents." + }, + { + "type": "text", + "bbox": [ + 0.143, + 0.842, + 0.367, + 0.857 + ], + "angle": 0, + "content": "- Your complete solution." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.86, + 0.482, + 0.875 + ], + "angle": 0, + "content": "In-text Citations and A Reference List." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.878, + 0.686, + 0.896 + ], + "angle": 0, + "content": "AI Use Report (If used does not count toward the 25-page limit.)" + }, + { + "type": "list", + "bbox": [ + 0.142, + 0.805, + 0.686, + 0.896 + ], + "angle": 0, + "content": null + }, + { + "type": "footer", + "bbox": [ + 0.225, + 0.939, + 0.77, + 0.957 + ], + "angle": 0, + "content": "©2026 by COMAP | www.mathmodels.org | info@comap.org |" + } + ], + [ + { + "type": "text", + "bbox": [ + 0.111, + 0.091, + 0.881, + 0.214 + ], + "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. 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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." + } + ] + } + ], + "index": 10 + }, + { + "bbox": [ + 67, + 403, + 526, + 467 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 403, + 526, + 467 + ], + "spans": [ + { + "bbox": [ + 67, + 403, + 526, + 467 + ], + "type": "text", + "content": "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." + } + ] + } + ], + "index": 11 + }, + { + "bbox": [ + 69, + 489, + 147, + 503 + ], + "type": "title", + "angle": 0, + "lines": [ + { + "bbox": [ + 69, + 489, + 147, + 503 + ], + "spans": [ + { + "bbox": [ + 69, + 489, + 147, + 503 + ], + "type": "text", + "content": "Requirements:" + } + ] + } + ], + "index": 12 + }, + { + "bbox": [ + 87, + 510, + 530, + 589 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 87, + 510, + 530, + 589 + ], + "spans": [ + { + "bbox": [ + 87, + 510, + 530, + 589 + ], + "type": "text", + "content": "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." + } + ] + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 606, + 537, + 716 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 104, + 606, + 537, + 716 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 537, + 716 + ], + "type": "text", + "content": "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." + } + ] + } + ], + "index": 14 + } + ], + "discarded_blocks": [ + { + "bbox": [ + 137, + 743, + 471, + 757 + ], + "type": "footer", + "angle": 0, + "lines": [ + { + "bbox": [ + 137, + 743, + 471, + 757 + ], + "spans": [ + { + "bbox": [ + 137, + 743, + 471, + 757 + ], + "type": "text", + "content": "©2026 by COMAP | www.mathmodels.org | www.mathmodels.org | info@comap.org |" + } + ] + } + ], + "index": 15 + } + ], + "page_size": [ + 612, + 792 + ], + "page_idx": 0 + }, + { + "para_blocks": [ + { + "bbox": [ + 86, + 72, + 523, + 133 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 72, + 523, + 133 + ], + "spans": [ + { + "bbox": [ + 86, + 72, + 523, + 133 + ], + "type": "text", + "content": "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." + } + ] + } + ], + "index": 0 + }, + { + "bbox": [ + 123, + 135, + 518, + 198 + ], + "type": "list", + "angle": 0, + "index": 3, + "blocks": [ + { + "bbox": [ + 123, + 135, + 518, + 165 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 123, + 135, + 518, + 165 + ], + "spans": [ + { + "bbox": [ + 123, + 135, + 518, + 165 + ], + "type": "text", + "content": "○ Show how your model explains differences in these outcomes and identify the specific drivers of rapid battery drain in each case." + } + ] + } + ], + "index": 1 + }, + { + "bbox": [ + 123, + 167, + 514, + 198 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 123, + 167, + 514, + 198 + ], + "spans": [ + { + "bbox": [ + 123, + 167, + 514, + 198 + ], + "type": "text", + "content": "- Which activities or conditions produce the greatest reductions in battery life? Which ones change the model surprisingly little?" + } + ] + } + ], + "index": 2 + } + ], + "sub_type": "text" + }, + { + "bbox": [ + 86, + 215, + 538, + 246 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 215, + 538, + 246 + ], + "spans": [ + { + "bbox": [ + 86, + 215, + 538, + 246 + ], + "type": "text", + "content": "3. Sensitivity and Assumptions: Examine how your predictions vary after making changes in your modeling assumptions, parameter values, and fluctuations in usage patterns." + } + ] + } + ], + "index": 4 + }, + { + "bbox": [ + 86, + 262, + 535, + 373 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 262, + 535, + 373 + ], + "spans": [ + { + "bbox": [ + 86, + 262, + 535, + 373 + ], + "type": "text", + "content": "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." + } + ] + } + ], + "index": 5 + }, + { + "bbox": [ + 69, + 389, + 217, + 403 + ], + "type": "title", + "angle": 0, + "lines": [ + { + "bbox": [ + 69, + 389, + 217, + 403 + ], + "spans": [ + { + "bbox": [ + 69, + 389, + 217, + 403 + ], + "type": "text", + "content": "Your report should present:" + } + ] + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 411, + 454, + 503 + ], + "type": "list", + "angle": 0, + "index": 12, + "blocks": [ + { + "bbox": [ + 105, + 411, + 412, + 426 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 105, + 411, + 412, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 412, + 426 + ], + "type": "text", + "content": "- A clear description of your model and governing equations." + } + ] + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 427, + 410, + 441 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 105, + 427, + 410, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 410, + 441 + ], + "type": "text", + "content": "- The assumptions and rationale behind your design choices." + } + ] + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 443, + 380, + 456 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 105, + 443, + 380, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 380, + 456 + ], + "type": "text", + "content": "Parameter estimation methods and validation results." + } + ] + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 459, + 426, + 472 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 105, + 459, + 426, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 426, + 472 + ], + "type": "text", + "content": "- A discussion of strengths, limitations, and possible extensions." + } + ] + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 475, + 454, + 503 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 105, + 475, + 454, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 454, + 503 + ], + "type": "text", + "content": "- An executive-style summary highlighting main results, insights, and recommendations." + } + ] + } + ], + "index": 11 + } + ], + "sub_type": "text" + }, + { + "bbox": [ + 67, + 522, + 533, + 601 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 522, + 533, + 601 + ], + "spans": [ + { + "bbox": [ + 67, + 522, + 533, + 601 + ], + "type": "text", + "content": "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." + } + ] + } + ], + "index": 13 + }, + { + "bbox": [ + 69, + 621, + 389, + 635 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 69, + 621, + 389, + 635 + ], + "spans": [ + { + "bbox": [ + 69, + 621, + 389, + 635 + ], + "type": "text", + "content": "Your PDF solution of no more than 25 total pages should include:" + } + ] + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 637, + 419, + 709 + ], + "type": "list", + "angle": 0, + "index": 20, + "blocks": [ + { + "bbox": [ + 86, + 637, + 236, + 650 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 637, + 236, + 650 + ], + "spans": [ + { + "bbox": [ + 86, + 637, + 236, + 650 + ], + "type": "text", + "content": "One-page Summary Sheet." + } + ] + } + ], + "index": 15 + }, + { + "bbox": [ + 87, + 651, + 197, + 663 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 87, + 651, + 197, + 663 + ], + "spans": [ + { + "bbox": [ + 87, + 651, + 197, + 663 + ], + "type": "text", + "content": "Table of Contents." + } + ] + } + ], + "index": 16 + }, + { + "bbox": [ + 87, + 666, + 224, + 678 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 87, + 666, + 224, + 678 + ], + "spans": [ + { + "bbox": [ + 87, + 666, + 224, + 678 + ], + "type": "text", + "content": "- Your complete solution." + } + ] + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 681, + 294, + 693 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 681, + 294, + 693 + ], + "spans": [ + { + "bbox": [ + 86, + 681, + 294, + 693 + ], + "type": "text", + "content": "In-text Citations and A Reference List." + } + ] + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 695, + 419, + 709 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 86, + 695, + 419, + 709 + ], + "spans": [ + { + "bbox": [ + 86, + 695, + 419, + 709 + ], + "type": "text", + "content": "AI Use Report (If used does not count toward the 25-page limit.)" + } + ] + } + ], + "index": 19 + } + ], + "sub_type": "text" + } + ], + "discarded_blocks": [ + { + "bbox": [ + 137, + 743, + 471, + 757 + ], + "type": "footer", + "angle": 0, + "lines": [ + { + "bbox": [ + 137, + 743, + 471, + 757 + ], + "spans": [ + { + "bbox": [ + 137, + 743, + 471, + 757 + ], + "type": "text", + "content": "©2026 by COMAP | www.mathmodels.org | info@comap.org |" + } + ] + } + ], + "index": 21 + } + ], + "page_size": [ + 612, + 792 + ], + "page_idx": 1 + }, + { + "para_blocks": [ + { + "bbox": [ + 67, + 72, + 539, + 169 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 72, + 539, + 169 + ], + "spans": [ + { + "bbox": [ + 67, + 72, + 539, + 169 + ], + "type": "text", + "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." + } + ] + } + ], + "index": 0 + }, + { + "bbox": [ + 69, + 185, + 118, + 199 + ], + "type": "title", + "angle": 0, + "lines": [ + { + "bbox": [ + 69, + 185, + 118, + 199 + ], + "spans": [ + { + "bbox": [ + 69, + 185, + 118, + 199 + ], + "type": "text", + "content": "Glossary" + } + ] + } + ], + "index": 1 + }, + { + "bbox": [ + 67, + 216, + 533, + 246 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 216, + 533, + 246 + ], + "spans": [ + { + "bbox": [ + 67, + 216, + 533, + 246 + ], + "type": "text", + "content": "Smartphone: is a mobile device that combines the functionality of a traditional cell phone with advanced computing capabilities." + } + ] + } + ], + "index": 2 + }, + { + "bbox": [ + 67, + 264, + 535, + 293 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 264, + 535, + 293 + ], + "spans": [ + { + "bbox": [ + 67, + 264, + 535, + 293 + ], + "type": "text", + "content": "Power Consumption: the rate at which a device uses electrical energy from its battery or power source." + } + ] + } + ], + "index": 3 + }, + { + "bbox": [ + 69, + 312, + 516, + 326 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 69, + 312, + 516, + 326 + ], + "spans": [ + { + "bbox": [ + 69, + 312, + 516, + 326 + ], + "type": "text", + "content": "Processor Load: the actual amount of work being done by the processor at a given moment." + } + ] + } + ], + "index": 4 + }, + { + "bbox": [ + 67, + 343, + 539, + 374 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 343, + 539, + 374 + ], + "spans": [ + { + "bbox": [ + 67, + 343, + 539, + 374 + ], + "type": "text", + "content": "State of Charge (SOC): a measure of how much energy remains in a battery compared to its full capacity, expressed as a percentage." + } + ] + } + ], + "index": 5 + }, + { + "bbox": [ + 67, + 391, + 496, + 422 + ], + "type": "text", + "angle": 0, + "lines": [ + { + "bbox": [ + 67, + 391, + 496, + 422 + ], + "spans": [ + { + "bbox": [ + 67, + 391, + 496, + 422 + ], + "type": "text", + "content": "Time-to-Empty: the estimated amount of time remaining before a battery is completely discharged." + } + ] + } + ], + "index": 6 + } + ], + "discarded_blocks": [ + { + "bbox": [ + 138, + 743, + 471, + 757 + ], + "type": "footer", + "angle": 0, + "lines": [ + { + "bbox": [ + 138, + 743, + 471, + 757 + ], + "spans": [ + { + "bbox": [ + 138, + 743, + 471, + 757 + ], + "type": "text", + "content": "©2026 by COMAP | www.mathmodels.org | info@comap.org |" + } + ] + } + ], + "index": 7 + } + ], + "page_size": [ + 612, + 792 + ], + "page_idx": 2 + } + ], + "_backend": "vlm", + "_version_name": "2.6.4" +} \ No newline at end of file diff --git a/2026_MCM_Problem_B.pdf-0c4ef00c-1c64-43ad-849c-d9c315561fbf/ffbd3f1f-bbe1-46f9-b0b1-122d0cf6051c_content_list.json b/2026_MCM_Problem_B.pdf-0c4ef00c-1c64-43ad-849c-d9c315561fbf/ffbd3f1f-bbe1-46f9-b0b1-122d0cf6051c_content_list.json new file mode 100644 index 0000000..4c26ca7 --- /dev/null +++ b/2026_MCM_Problem_B.pdf-0c4ef00c-1c64-43ad-849c-d9c315561fbf/ffbd3f1f-bbe1-46f9-b0b1-122d0cf6051c_content_list.json @@ -0,0 +1,336 @@ +[ + { + "type": "text", + "text": "2026 MCM", + "text_level": 1, + "bbox": [ + 450, + 90, + 547, + 106 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Problem B: Creating a Moon Colony Using a Space Elevator System", + "text_level": 1, + "bbox": [ + 210, + 111, + 785, + 128 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/d56c8f8a71b712570aacdb699e2e19450e8b1a708748d394f92cbee59747b814.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 292, + 138, + 702, + 313 + ], + "page_idx": 0 + }, + { + "type": "text", + "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.", + "bbox": [ + 114, + 327, + 880, + 507 + ], + "page_idx": 0 + }, + { + "type": "text", + "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.", + "bbox": [ + 114, + 518, + 867, + 638 + ], + "page_idx": 0 + }, + { + "type": "text", + "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.", + "bbox": [ + 114, + 648, + 870, + 808 + ], + "page_idx": 0 + }, + { + "type": "text", + "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).", + "bbox": [ + 114, + 824, + 867, + 895 + ], + "page_idx": 0 + }, + { + "type": "footer", + "text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org", + "bbox": [ + 230, + 939, + 766, + 954 + ], + "page_idx": 0 + }, + { + "type": "text", + "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.", + "bbox": [ + 109, + 90, + 872, + 196 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Your Task:", + "text_level": 1, + "bbox": [ + 112, + 213, + 210, + 231 + ], + "page_idx": 1 + }, + { + "type": "text", + "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.", + "bbox": [ + 109, + 242, + 880, + 321 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Your model should include:", + "text_level": 1, + "bbox": [ + 112, + 334, + 341, + 349 + ], + "page_idx": 1 + }, + { + "type": "text", + "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;", + "bbox": [ + 111, + 364, + 883, + 402 + ], + "page_idx": 1 + }, + { + "type": "list", + "sub_type": "text", + "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." + ], + "bbox": [ + 140, + 415, + 870, + 488 + ], + "page_idx": 1 + }, + { + "type": "list", + "sub_type": "text", + "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." + ], + "bbox": [ + 111, + 505, + 874, + 734 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Your PDF solution of no more than 25 total pages should include:", + "bbox": [ + 112, + 747, + 635, + 765 + ], + "page_idx": 1 + }, + { + "type": "list", + "sub_type": "text", + "list_items": [ + "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.)" + ], + "bbox": [ + 140, + 767, + 684, + 875 + ], + "page_idx": 1 + }, + { + "type": "footer", + "text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org", + "bbox": [ + 225, + 938, + 767, + 955 + ], + "page_idx": 1 + }, + { + "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": [ + 109, + 90, + 880, + 213 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Glossary", + "text_level": 1, + "bbox": [ + 112, + 229, + 194, + 250 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Space Elevator System is comprised of three Galactic Harbours plus additional support facilities.", + "bbox": [ + 109, + 265, + 815, + 299 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Galactic Harbour is comprised of two apex anchors each connected by two tethers to a single Earth Port.", + "bbox": [ + 109, + 316, + 861, + 351 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Earth Port is the location on the Earth that provides surface support for the Galactic Harbour.", + "bbox": [ + 112, + 369, + 857, + 386 + ], + "page_idx": 2 + }, + { + "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.", + "bbox": [ + 109, + 404, + 874, + 439 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Apex Anchor is the counterweight in space at the end of the $100,000\\mathrm{km}$ tether.", + "bbox": [ + 112, + 455, + 743, + 474 + ], + "page_idx": 2 + }, + { + "type": "text", + "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.", + "bbox": [ + 109, + 491, + 883, + 544 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Moon Colony is a habitat on the moon with the capacity to support 100,000-persons.", + "bbox": [ + 112, + 560, + 787, + 579 + ], + "page_idx": 2 + }, + { + "type": "footer", + "text": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org |", + "bbox": [ + 227, + 938, + 767, + 955 + ], + "page_idx": 2 + } +] \ No newline at end of file diff --git a/2026_MCM_Problem_B.pdf-0c4ef00c-1c64-43ad-849c-d9c315561fbf/ffbd3f1f-bbe1-46f9-b0b1-122d0cf6051c_model.json b/2026_MCM_Problem_B.pdf-0c4ef00c-1c64-43ad-849c-d9c315561fbf/ffbd3f1f-bbe1-46f9-b0b1-122d0cf6051c_model.json new file mode 100644 index 0000000..b86d7e5 --- /dev/null +++ b/2026_MCM_Problem_B.pdf-0c4ef00c-1c64-43ad-849c-d9c315561fbf/ffbd3f1f-bbe1-46f9-b0b1-122d0cf6051c_model.json @@ -0,0 +1,459 @@ +[ + [ + { + "type": "title", + "bbox": [ + 0.451, + 0.092, + 0.548, + 0.107 + ], + "angle": 0, + "content": "2026 MCM" + }, + { + "type": "title", + "bbox": [ + 0.212, + 0.112, + 0.786, + 0.129 + ], + "angle": 0, + "content": "Problem B: Creating a Moon Colony Using a Space Elevator System" + }, + { + "type": "image", + "bbox": [ + 0.294, + 0.14, + 0.703, + 0.314 + ], + "angle": 0, + "content": null + }, + { + "type": "text", + "bbox": [ + 0.115, + 0.328, + 0.882, + 0.508 + ], + "angle": 0, + "content": "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." + }, + { + "type": "text", + "bbox": [ + 0.116, + 0.52, + 0.869, + 0.64 + ], + "angle": 0, + "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." + }, + { + "type": "text", + "bbox": [ + 0.116, + 0.65, + 0.872, + 0.809 + ], + "angle": 0, + "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." + }, + { + "type": "text", + "bbox": [ + 0.116, + 0.825, + 0.869, + 0.896 + ], + "angle": 0, + "content": "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)." + }, + { + "type": "footer", + "bbox": [ + 0.231, + 0.94, + 0.767, + 0.955 + ], + "angle": 0, + "content": "©2026 by COMAP | www.comap.org | www.mathmodels.org | info@comap.org" + } + ], + [ + { + "type": "text", + "bbox": [ + 0.111, + 0.091, + 0.874, + 0.198 + ], + "angle": 0, + "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." + }, + { + "type": "title", + "bbox": [ + 0.113, + 0.214, + 0.212, + 0.232 + ], + "angle": 0, + "content": "Your Task:" + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.243, + 0.882, + 0.322 + ], + "angle": 0, + "content": "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." + }, + { + "type": "title", + "bbox": [ + 0.114, + 0.335, + 0.342, + 0.351 + ], + "angle": 0, + "content": "Your model should include:" + }, + { + "type": "text", + "bbox": [ + 0.112, + 0.365, + 0.884, + 0.403 + ], + "angle": 0, + "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;" + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.416, + 0.687, + 0.43 + ], + "angle": 0, + "content": "a. using the Space Elevator System's three Galactic Harbor's alone," + }, + { + "type": "text", + "bbox": [ + 0.141, + 0.435, + 0.872, + 0.47 + ], + "angle": 0, + "content": "b. traditional rocket launches from existing bases alone (you may choose which facilities to use), or," + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.476, + 0.481, + 0.489 + ], + "angle": 0, + "content": "c. some combination of the two methods." + }, + { + "type": "list", + "bbox": [ + 0.141, + 0.416, + 0.872, + 0.489 + ], + "angle": 0, + "content": null + }, + { + "type": "text", + "bbox": [ + 0.112, + 0.506, + 0.852, + 0.545 + ], + "angle": 0, + "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.).?" + }, + { + "type": "text", + "bbox": [ + 0.112, + 0.556, + 0.875, + 0.613 + ], + "angle": 0, + "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." + }, + { + "type": "text", + "bbox": [ + 0.112, + 0.627, + 0.822, + 0.685 + ], + "angle": 0, + "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?" + }, + { + "type": "text", + "bbox": [ + 0.112, + 0.698, + 0.841, + 0.736 + ], + "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." + }, + { + "type": "list", + "bbox": [ + 0.112, + 0.506, + 0.875, + 0.736 + ], + "angle": 0, + "content": null + }, + { + "type": "text", + "bbox": [ + 0.113, + 0.748, + 0.637, + 0.766 + ], + "angle": 0, + "content": "Your PDF solution of no more than 25 total pages should include:" + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.768, + 0.387, + 0.782 + ], + "angle": 0, + "content": "One-page Summary Sheet." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.786, + 0.321, + 0.799 + ], + "angle": 0, + "content": "Table of Contents." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.805, + 0.365, + 0.818 + ], + "angle": 0, + "content": "- Your complete solution." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.823, + 0.436, + 0.838 + ], + "angle": 0, + "content": "One-page letter to MCM Agency" + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.842, + 0.297, + 0.854 + ], + "angle": 0, + "content": "References list." + }, + { + "type": "text", + "bbox": [ + 0.142, + 0.859, + 0.685, + 0.875 + ], + "angle": 0, + "content": "AI Use Report (If used does not count toward the 25-page limit.)" + }, + { + "type": "list", + "bbox": [ + 0.142, + 0.768, + 0.685, + 0.875 + ], + "angle": 0, + "content": null + }, + { + "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" + } + ], + [ + { + "type": "text", + "bbox": [ + 0.111, + 0.091, + 0.881, + 0.214 + ], + "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": "title", + "bbox": [ + 0.113, + 0.231, + 0.196, + 0.25 + ], + "angle": 0, + "content": "Glossary" + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.266, + 0.816, + 0.3 + ], + "angle": 0, + "content": "Space Elevator System is comprised of three Galactic Harbours plus additional support facilities." + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.318, + 0.862, + 0.352 + ], + "angle": 0, + "content": "Galactic Harbour is comprised of two apex anchors each connected by two tethers to a single Earth Port." + }, + { + "type": "text", + "bbox": [ + 0.113, + 0.37, + 0.859, + 0.387 + ], + "angle": 0, + "content": "Earth Port is the location on the Earth that provides surface support for the Galactic Harbour." + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.405, + 0.875, + 0.44 + ], + "angle": 0, + "content": "Tethers are \\(100,000\\mathrm{km}\\) long graphene material that links the Earth port and apex anchors in the Space Elevator System." + }, + { + "type": "text", + "bbox": [ + 0.113, + 0.457, + 0.744, + 0.476 + ], + "angle": 0, + "content": "Apex Anchor is the counterweight in space at the end of the \\(100,000\\mathrm{km}\\) tether." + }, + { + "type": "text", + "bbox": [ + 0.111, + 0.492, + 0.885, + 0.545 + ], + "angle": 0, + "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." + }, + { + "type": "text", + "bbox": [ + 0.113, + 0.561, + 0.788, + 0.58 + ], + "angle": 0, + "content": "Moon Colony is a habitat on the moon with the capacity to support 100,000-persons." + }, + { + "type": "footer", + "bbox": [ + 0.228, + 0.939, + 0.769, + 0.957 + ], + "angle": 0, + "content": "©2026 by COMAP | 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