2025MCM – 美国大学生数学建模竞赛

2025 MCM Problem C: Models for Olympic Medal Tables

During the 2024 Paris Summer Olympics, in addition to watching various individual events, fans closely followed the overall “medal table” for each country. The final results (Table 1) showed that the United States had the largest total number of medals (126). China and the United States tied for the first place in the gold medal standings, both with 40 gold medals. The host country, France, ranked 5th in the gold medal count (16) but 4th in terms of the total medal count. Great Britain, with 14 gold medals ranking 7th, was 3rd in the total medal count.

Table 1: Final Medal Table of the 2024 Paris Olympics – Top 7 Countries in Gold Medal Count [1]

The rankings at the top of the medal table always attract much attention, but the medal counts of other countries are also of great significance. For example, Albania (2 medals), Cabo Verde, Dominica, and Saint Lucia (2 medals each) won their first Olympic medals at the Paris Olympics. Dominica and Saint Lucia each also earned a gold medal. More than 60 countries still haven’t won an Olympic medal so far.

Predictions of the final medal counts are often made. Usually, they are not based on historical medal counts but are made closer to the start of an upcoming Olympics when the information of current athletes scheduled to compete is available (for example: https://www.nielsen.com/news – center/2024/virtual – medal – table – forecast/).

Data of medal tables for all summer Olympics, host countries, as well as the number of Olympic events at each games broken down by sport are provided. In addition, data of all individual Olympic competitors, including their sports and results (medal types or no medals), are also given. Your models and data analysis must only use the provided datasets. You can use additional resources to provide background, context, or assistance in interpreting results (make sure to document the sources). Specifically, use the provided data to:

• Develop a model for medal counts for each country (at least for gold medals and total medals). Include estimates of the uncertainty/precision of your model predictions and measures of how well the model performs.
  • Based on your model, what are your projections for the medal table in the 2028 Los Angeles Summer Olympics in the United States? Include prediction intervals for all results. Which countries do you think are most likely to improve? Which countries will perform worse than in 2024?
  • Your model should cover countries that haven’t won medals yet. What is your projection for the number of countries that will win their first medal in the next Olympics? How confident are you about this estimate?
  • Your model should also take into account the events (number and types) at a given Olympics. Explore the relationship between the events and the number of medals countries win. Which sports are most important for different countries? Why? How do the events selected by the host country affect the results?

Athletes may compete for different countries, but it’s not easy for them to change due to citizenship requirements. Coaches, however, can move from one country to another easily as they don’t need to be citizens to coach. Therefore, there may be a “great coach” effect. Two possible examples are Lang Ping [2], who coached the volleyball teams of both the US and China to championships, and the sometimes – controversial gymnastics coach, Béla Károlyi [3], who coached the Romanian and then the US women’s gymnastics teams with great success. Examine the data to find evidence of changes that might be attributed to the “great coach” effect. Estimate how much this effect contributes to the medal counts. Select three countries, identify the sports in which they should consider hiring a “great” coach, and estimate the impact.

What other original insights about Olympic medal counts does your model reveal? Explain how these insights can provide references for national Olympic committees.

Your PDF solution, with a maximum of 25 pages in total, should include:

• A one – page Summary Sheet.
• A Table of Contents.
• Your complete solution.
• A References list.
• An AI Use Report (If used, it doesn’t count towards the 25 – page limit.)

Note: There is no specific minimum page length requirement for a complete MCM submission. You can use up to 25 pages for all your solution work and any additional information you want to include (such as drawings, diagrams, calculations, tables). Partial solutions are acceptable. We allow the cautious use of AI, such as ChatGPT, although it’s not necessary to solve this problem. If you choose to use generative AI, you must follow COMAP’s AI use policy. This will require you to add an AI use report at the end of your PDF solution file, which doesn’t count towards the 25 – page limit of your solution.

New MCM/ICM: Online Submission Process

This article aims to assist and guide students and advisors participating in HiMCM/MidMCM. COMAP introduces the new online submission process using the new online submission page https://forms.comap.org/241335097294056 in this article. You need your team’s control number, advisor ID number, and the problem you’ve chosen to complete the submission.

Data Files

2025_Problem_C_Data.zip: This zip file contains all 5 data files listed below.

• data_dictionary.csv – Database descriptions with examples.
• summerOly_athletes.csv – Information of all competitors including their sports, years of participation, and results (medal types or no medals).
• summerOly_medal_counts.csv – Complete country medal – count tables for all summer Olympics from 1896 to 2024.
• summerOly_hosts.csv – List of host countries for all summer Olympics from 1896 to 2032.
• summerOly_programs.csv – Counts of the number of events by sport/discipline and the total for all summer Olympics from 1896 to 2032.

Data such as country designations are recorded by the International Olympic Committee (IOC) on their Olympics.com website during a specific Olympics. So, the designations in the dataset may change. Like all data, there may be recording anomalies. For example, in the athlete dataset, for sports like tennis, table tennis, and beach volleyball, the “Team” information sometimes includes more details than just the country. For instance, Germany – 1 represented the first of two beach volleyball teams from Germany in the 2000 Olympics. Decisions and assumptions about how to handle the data are crucial parts of the modeling process.

Glossary

• International Olympic Committee (IOC): It is the international, non – governmental sports – governing body of the Olympic Games and the Olympic Movement. The IOC is well – known for organizing the Summer and Winter Olympics.
• Programme of the Olympic Games: It is the program of all sports competitions established by the IOC for each edition of the Olympic Games.
• SDE: Sport, Discipline, or Event.
• Sport: The IOC defines an Olympic sport as a discipline governed by a single international sports federation (IF). A single sport may contain one or more disciplines, and each discipline is the focus of one or more events.
• Discipline: A branch of a sport that includes one or more events.
• Event: A competition within a discipline that results in a ranking and awards (such as medals).

Take the 2024 Paris Olympics as an example to illustrate the relationship between sport, discipline, and event: World Aquatics is the IF that governs the sport of aquatics. In the sport of aquatics, there are multiple disciplines, such as artistic swimming, diving, marathon swimming, swimming, and water polo. In the discipline of diving, there are 8 medal events: men’s and women’s individual 3m springboard, men’s and women’s individual 10m platform, men’s and women’s synchronized 3m springboard, and men’s and women’s synchronized 10m platform.

References

[1] Olympics.com, https://olympics.com/en/paris – 2024/medals
[2] Olympics.com Biography, Lang Ping, https://olympics.com/en/athletes/ping – lang
[3] USA Gymnastics Hall of Fame, https://usagym.org/halloffame/inductee/coaching – team – bela – martha – karolyi/

Disclaimer

COMAP is a non – profit organization dedicated to improving mathematics education, especially focusing on enhancing students’ proficiency in mathematical modeling. This contest problem quotes materials from the International Olympic Committee (IOC). We acknowledge and respect the IOC’s ownership of these materials. The use here is solely for educational and non – commercial purposes to enrich the learning experiences of participants. This content is not endorsed by or affiliated with the IOC.

2025 年数学建模竞赛（MCM）C 题：奥运会奖牌榜模型

在 2024 年巴黎夏季奥运会期间，除了观看各项单项赛事，粉丝们还密切关注着每个国家的总 “奖牌榜” 情况。最终结果（表 1）显示，美国获得的奖牌总数最多，达 126 枚，中国和美国在金牌榜上并列第一，均获得 40 枚金牌 。主办国法国在金牌数排名中位列第 5（共 16 枚金牌），但在奖牌总数排名中位列第 4；英国以 14 枚金牌排在第 7 位，不过在奖牌总数排名中位居第 3。

表 1：2024 年巴黎奥运会最终奖牌榜 —— 金牌数前 7 的国家 [1]

奖牌榜前列的排名总是备受关注，但其他国家的奖牌数也同样受到重视。例如，阿尔巴尼亚（获得 2 枚奖牌）、佛得角、多米尼克和圣卢西亚（均获得 2 枚奖牌）在巴黎奥运会上赢得了本国的首枚奥运奖牌。多米尼克和圣卢西亚在这届奥运会上还各自斩获了一枚金牌。目前，仍有 60 多个国家尚未赢得过奥运奖牌。

人们通常会对奥运会最终的奖牌数进行预测，但这类预测一般并非基于历史奖牌数据，而是在临近奥运会开幕、确定了计划参赛的现役运动员信息后才进行（例如：https://www.nielsen.com/news – center/2024/virtual – medal – table – forecast/ ）。

此次提供了所有夏季奥运会的奖牌榜数据、主办国信息，以及各届夏季奥运会按运动项目细分的赛事数量。此外，还提供了所有奥运会个人参赛者的运动项目和比赛结果（获得何种奖牌或未获奖）数据。你的模型和数据分析必须仅使用提供的数据集。你可以利用其他资源来提供背景信息、上下文内容，或辅助解释结果（务必注明资料来源）。具体而言，需使用提供的数据来：

• 为每个国家构建奖牌数模型（至少针对金牌数和奖牌总数）。给出模型预测的不确定性 / 精度估计，以及评估模型性能的指标。
  • 根据你的模型，对 2028 年美国洛杉矶夏季奥运会的奖牌榜有何预测？给出所有预测结果的置信区间。你认为哪些国家最有可能取得进步？哪些国家会比 2024 年表现更差？
  • 你的模型应涵盖尚未获得过奖牌的国家；你预测在下一届奥运会中有多少个国家将赢得首枚奖牌？你对这一预测的把握有多大？
  • 你的模型还应考虑特定奥运会的赛事（数量和类型）。探究赛事与各国获得奖牌数量之间的关系。哪些运动项目对不同国家最为重要？原因是什么？主办国选定的赛事对比赛结果有何影响？

运动员更换代表国家并非易事，因为这受到公民身份要求的限制。然而，教练更换执教国家则相对容易，因为他们执教无需拥有该国国籍。因此，可能存在 “优秀教练效应”。例如，郎平 [2] 曾分别执教美国和中国的排球队并带领队伍夺冠；还有体操教练贝拉・卡罗利 [3]，他先后执教罗马尼亚和美国女子体操队并取得显著成就，不过其执教方式有时也颇具争议。请研究数据，寻找可能归因于 “优秀教练效应” 的变化证据。估计这种效应在奖牌数中所占的比重。选择三个国家，确定这些国家在哪些运动项目上应考虑聘请 “优秀” 教练，并预估其带来的影响。

你的模型还揭示了哪些关于奥运奖牌数的独到见解？解释这些见解如何为各国奥委会提供参考。

你提交的 PDF 解决方案总页数不得超过 25 页，应包含：

• 1 页摘要表
• 目录
• 完整的解决方案
• 参考文献列表
• AI 使用报告（若使用了 AI，此报告不计入 25 页的总页数限制）

注意：完整的 MCM 提交作品没有具体的最低页数要求。你最多可使用 25 页来完成解决方案以及包含任何你想添加的额外信息（例如：绘图、图表、计算过程、表格等）。部分解决方案也可接受。我们允许谨慎使用像 ChatGPT 这样的人工智能工具，不过解决此问题并非必须使用。若你选择使用生成式人工智能，必须遵循 COMAP 的人工智能使用政策。这意味着你需要在 PDF 解决方案文件末尾添加一份 AI 使用报告，该报告不计入解决方案的 25 页总页数限制。

新 MCM/ICM：在线提交流程

本文旨在为参加高中数学建模竞赛（HiMCM）/ 中学数学建模竞赛（MidMCM）的学生和指导老师提供帮助与指导。在本文中，COMAP 介绍了使用新在线提交页面（https://forms.comap.org/241335097294056）进行在线提交的新流程。你需要用到团队控制编号、指导老师 ID 编号以及所选题目，才能完成提交。

数据文件

2025_Problem_C_Data.zip：此压缩文件包含以下列出的全部 5 个数据文件。

• data_dictionary.csv：带有示例的数据库描述
• summerOly_athletes.csv：所有参赛者的运动项目、参赛年份和比赛结果（获得何种奖牌或未获奖）
• summerOly_medal_counts.csv：1896 年至 2024 年所有夏季奥运会完整的国家奖牌数表格
• summerOly_hosts.csv：1896 年至 2032 年所有夏季奥运会的主办国列表
• summerOly_programs.csv：1896 年至 2032 年所有夏季奥运会按运动项目 / 分项统计的赛事数量及总数

数据（如国家名称）由国际奥委会（IOC）在特定奥运会举办时记录在其Olympics.com网站上。因此，数据集中的国家名称等信息可能会发生变化。和所有数据一样，这些数据也可能存在记录异常的情况。例如，在运动员数据集中，对于网球、乒乓球、沙滩排球等项目，“团队” 信息有时包含比国家更详细的内容。以 2000 年奥运会为例，Germany – 1 代表德国两支沙滩排球队中的第一支。在建模过程中，如何处理这些数据的决策和假设至关重要。

术语表

• 国际奥委会（IOC）：是奥运会和奥林匹克运动的国际非政府体育管理机构。IOC 最为人熟知的职责是组织夏季和冬季奥运会。
• 奥运会项目：是 IOC 为每届奥运会制定的所有体育比赛项目。
• SDE：运动项目、分项或赛事
• 运动项目：IOC 将奥运会运动项目定义为由单一国际体育联合会（IF）管理的一个分项。一个运动项目可能包含一个或多个分项，每个分项是一个或多个赛事的核心。
• 分项：是一项运动的分支，包含一个或多个赛事。
• 赛事：是分项内的一场比赛，会产生排名并颁发奖项（如奖牌）。

以 2024 年巴黎奥运会为例，说明运动项目、分项和赛事之间的关系：世界游泳联合会（World Aquatics）是管理游泳运动的国际体育联合会。在游泳运动项目中，包含多个分项，如艺术游泳、跳水、马拉松游泳、游泳和水球。在跳水分项中，设有 8 个奖牌赛事，分别为男子和女子个人 3 米跳板、男子和女子个人 10 米跳台、男子和女子双人 3 米跳板、男子和女子双人 10 米跳台。

参考文献

[1] Olympics.com, https://olympics.com/en/paris – 2024/medals
[2] Olympics.com Biography, Lang Ping, https://olympics.com/en/athletes/ping – lang
[3] USA Gymnastics Hall of Fame, https://usagym.org/halloffame/inductee/coaching – team – bela – martha – karolyi/

免责声明

COMAP 是一个非营利组织，致力于改善数学教育，尤其注重提升学生的数学建模能力。本竞赛题目引用了国际奥委会（IOC）的材料。我们承认并尊重 IOC 对这些材料的所有权，在此使用仅出于教育、非商业目的，旨在丰富参与者的学习体验。本内容未得到 IOC 的认可，也与 IOC 没有任何关联。