2025年美国大学生数学建模竞赛C题中英版

中文赛题：奥运奖牌榜模型

在 2024 年巴黎举行的最新一届夏季奥运会上，除了观看个人项目外，粉丝们还关注了每个国家的整体“奖牌榜”。最终结果（表 1）显示，美国的总奖牌数最多（126 枚），中国和美国并列第一金牌榜首位（40 枚）。东道主法国在金牌数（16 枚）排名中位居第五，但在奖牌总数方面排名第四，而英国以 14 枚金牌排名第七，在奖牌总数方面排名第三。

排行榜顶端的排名总是受到密切关注，但其他国家的奖牌数往往也同样受到重视。例如，阿尔巴尼亚（2 枚奖牌）、佛得角、多米尼加和圣卢西亚（2 枚奖牌）在巴黎奥运会上赢得了本国首枚奥运奖牌。多米尼加和圣卢西亚也分别在这届奥运会上获得了一枚金牌。仍有 60 多个国家尚未赢得奥运奖牌。人们通常会预测最终的奖牌数，但通常不是基于历史奖牌数，而是在即将举行的奥运会开始前，届时将公布当前计划参赛的运动员（例如：https://www.nielsen.com/news-center/2024/virtual-medal-table-forecast/ ）。

提供了所有夏季奥运会奖牌榜、主办国以及所有夏季奥运会各届奥运会按运动项目细分的奥运赛事数量数据。此外，还提供了所有个人奥运会参赛者及其赛事和结果（奖牌类型或无奖牌）的数据。您的模型和数据分析必须仅使用提供的数据集。您可以使用其他资源来提供背景和上下文或帮助解释结果（请务必记录来源）。具体来说，使用提供的数据来：

• 为每个国家（至少为金牌和奖牌总数）开发一个奖牌计数模型。包括模型预测的不确定性/精确度的估计值和模型表现的衡量标准。
  • 根据您的模型，您对 2028 年美国洛杉矶夏季奥运会奖牌榜的预测是什么？包括所有结果的预测区间。您认为哪些国家最有可能进步？哪些国家的表现会比 2024 年更差？
  • 您的模型应包括尚未获得奖牌的国家；您预测有多少国家将在下届奥运会上获得第一枚奖牌？您对这个估计的几率是多少？
  • 您的模型还应考虑特定奥运会的赛事（数量和类型）。探索赛事与国家获得的奖牌数之间的关系。哪些运动对各个国家最重要？为什么？本国选择的项目如何影响结果？
• 运动员可以代表不同的国家参赛，但由于公民身份要求，他们很难更换国家。然而，教练可以轻松地从一个国家转移到另一个国家，因为他们不需要是公民就可以执教。因此，有可能出现“伟大教练”效应。两个可能的例子包括郎平^[2]，他曾执教美国和中国的排球队，并夺得冠军；以及有时备受争议的体操教练贝拉·卡罗利^[3]，他先后执教罗马尼亚和美国女子队，取得了巨大的成功。检查数据，寻找可能因“伟大教练”效应而发生变化的证据。你估计这种影响对奖牌数的贡献有多大？选择三个国家，确定哪些运动项目应该考虑投资“伟大”教练，并估计这种影响。
• 您的模型还揭示了有关奥运奖牌数量的哪些其他原创见解？解释这些见解如何为国家奥委会提供信息。

您的 PDF 解决方案总页数不得超过25页，包括：

• 一页摘要页。
• 目录。
• 完整解决方案。
• 一封给交通官员的信（一页）。
• 参考文献列表。
• AI使用报告（如果使用不计入 25 页总限制）。

注意：完整的 MCM 提交文件没有特定的最低页数要求。您最多可使用 25 页来包含您的所有解决方案内容及其他补充信息（例如图纸、图表、计算和表格）。允许提交部分解决方案。我们允许谨慎使用 AI 工具（例如 ChatGPT），但完成本问题并不一定需要使用 AI 工具。如果您选择使用生成式 AI 工具，必须遵循 COMAP 的 AI使用政策。此政策要求您在解决方案文件的最后额外添加一份 AI 使用报告，不计入 25 页的总限制。

全新 MCM/ICM：在线提交流程

本文旨在协助和指导参加 MCM/ICM 的学生和顾问。在文章中，COMAP 提供了有关使用新在线提交页面的新在线提交流程的信息：

https://forms.comap.org/241335097294056

您需要团队的控制号、顾问 ID 号和问题选择才能完成提交。

数据文件

2025_Problem_C_Data.zip：此 zip 文件包含下面列出的所有 5 个数据文件。

• data_dictionary.csv – 带有示例的数据库描述
• summerOly_athletes.csv – 所有参赛者及其运动项目、年份和结果（奖牌类型或无）
• summerOly_medal_counts.csv – 1896 年至 2024 年所有夏季奥运会的完整国家奖牌统计表
• summerOly_hosts.csv – 1896 年至 2032 年所有夏季奥运会主办国名单
• summerOly_programs.csv – 1896 年至 2032 年所有夏季奥运会按运动项目/学科统计的赛事数量和总数

国家名称等数据由国际奥林匹克委员会(IOC)（在其 Olympics.com 网站上）在特定奥运会期间记录。因此，数据集中的名称可能会发生变化。与所有数据一样，记录可能存在异常。例如，请注意，在运动员数据集中，在某些情况下，对于网球、乒乓球、沙滩排球等运动，“球队”包含比国家更多的细节。例如，德国-1 将是 2000 年奥运会上来自德国的两支沙滩排球队中的第一支。关于如何处理数据的决策和假设是建模过程的重要组成部分。

术语表

国际奥林匹克委员会 (IOC)：是管理奥运会和奥林匹克运动的国际非政府体育机构。IOC 最为人熟知的是负责组织夏季和冬季奥运会的组织。

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、分项或赛事

• 运动项目：国际奥委会将奥林匹克运动项目定义为由单一国际体育联合会 (IF) 管理的项目。一项运动项目可能包含一个或多个项目，每个项目都是一个或多个赛事的焦点。
• 学科：包括一项或多项赛事的运动分支。
• 赛事：学科内的比赛，可产生排名和奖项（例如奖牌）。
• 2024 年巴黎奥运会项目中运动、学科和赛事之间关系的示例：
  • 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 - 男子和女子
  • 世界水上运动联合会是管理水上运动的国际体育联合会
  • 水上运动包括多个项目——花样游泳、跳水、马拉松游泳、游泳和水球。
  • 跳水项目包括八个奖牌项目：

免责声明

COMAP 是一家致力于改善数学教育的非盈利组织，重点是提高学生的数学建模能力。本次竞赛问题参考了国际奥林匹克委员会 (IOC) 的材料。我们承认并尊重 IOC 对这些材料的所有权，此处仅将其用于教育、非商业目的，以丰富参与者的学习体验。本内容不受 IOC 认可或隶属于 IOC。

Problem: Models for Olympic Medal Tables

In addition to watching individual events during the latest summer Olympic Games in Paris, 2024, fans followed the overall “medal table” for each country. The final results (Table 1) showed the United States with the most total medals (126), and China and the United States tied at the top of the standings for the first place Gold medals (40). The host country, France, was 5th in the Gold medal count (16) standings but 4th in terms of total medal count while Great Britain, 7th with 14 Gold medals finished 3rd in total medals.

The standings at the top of the table are always watched closely, but the medal counts for other countries are often just as valued. For example, Albania (2 medals), Cabo Verde, Dominica, and Saint Lucia (2 medals) won their nations' first Olympic medals at the Paris games. Dominica and Saint Lucia also each earned a Gold medal at these games. More than 60 countries have still yet to win an Olympic medal.

Predictions of the final medal counts are commonly made, but typically not based on historical medal counts but closer to the start of an upcoming Olympic games when current athletes scheduled to compete are known (for example: https://www.nielsen.com/news-center/2024/virtual-medal-table-forecast/).

Data is provided of medal tables for all summer Olympic games, host countries, as well as the number of Olympic events at each games broken down by sport for all summer Olympic games played. Additionally, data for all individual Olympic competitors with their sport and result(medal type, or no medal) is provided. Your models and data analysis must ONLY use the provided data sets. You may use additional resources to provide background and context or help with interpreting results (be sure to document the sources). Specifically, use the provided data to:

• Develop a model for medal counts for each country (for Gold and total medals at a minimum). Include estimates of the uncertainty/precision of your model predictions and measures of how well model performs.
  • Based on your model, what are your projections for the medal table in the Los Angeles, USA summer Olympics in 2028? Include prediction intervals for all results. Which countries do you believe are most likely to improve? Which will do worse than in 2024?
  • Your model should include countries that have yet to earn medals; what is your projection for how many will earn their first medal in the next Olympics? What sort of odds do you give to this estimate?
  • Your model should also consider the events (number and types) at a given Olympics. Explore the relationship between the events and how many medals countries earn. What sports are most important for various countries? Why? How do the events chosen by the home country impact results?
• Athletes may compete for different countries, but it is not a simple matter for them to change due to citizenship requirements. Coaches, however, can easily move from one country to another as they do not need to be citizens to coach. There is, therefore, the possibility of a “great coach” effect. Two possible examples of this include Lang Ping^[2], who coached volleyball teams from both the U.S. and China to championships, and the sometimes-controversial gymnastics coach, Béla Károlyi^[3], who coached Romania and then the U.S. women’s teams with great success. Examine the data for evidence of changes that might be due to a “great coach” effect. How much do you estimate such an effect contributes to medal counts? Choose three countries and identify sports where they should consider investing in a “great” coach and estimate that impact.
• What other original insight(s) about Olympic medal counts does your model reveal? Explain how these insight(s) can inform country Olympic committees.

Your PDF solution of no more than 25 total pages should include:

• One-page Summary Sheet.
• Table of Contents.
• Your complete solution.
• One-page letter to the transportation officials.
• References list.
• AI Use Report (If used does not count toward the 25-page limit.)

Note: There is no specific required minimum page length for a complete MCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.

NEW MCM/ICM: Online Submission Process

The purpose of this article is to assist and guide students and advisors participating in MCM/ICM. In the article, COMAP, provides information about the new online submission process using the new online submission page:

https://forms.comap.org/241335097294056

You will need your team's control number, advisor id number and your problem choice to complete your submission.

Data Files

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

• data_dictionary.csv – database descriptions with examples
• summerOly_athletes.csv – all competitors with their sport, year, and result (medal type or none)
• summerOly_medal_counts.csv – complete country medal count tables for all summer Olympics from 1896 to 2024
• summerOly_hosts.csv – list of host country for all summer Olympics from 1896 to 2032
• summerOly_programs.csv – counts of number of events by sport/discipline and 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) at the time of a given Olympics. Thus, designations may change in the data set. As with all data, there may be recording anomalies. Note, for example, in the athlete’s data set in some cases for sports like tennis, table tennis, beach volleyball, the “Team” includes more detail than just the country. For example, Germany-1 would be the first of two beach volleyball teams from Germany in the 2000 Olympics. Decisions and assumptions about how to handle the data are an important part of the modeling process.

Glossary

International Olympic Committee (IOC): is the international, non-governmental, sports governing body of the Olympic Games and the Olympic Movement. The IOC is best known as the organization responsible for organizing the Summer and Winter Olympics.

Programme: of the Olympic Games is the programme 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 that is governed by a single international sports federation (IF). A single sport may contain one or more disciplines, each of which 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 (e.g. medals).
• Example of the relationship between sport, discipline, and event in the Olympic programme from the 2024 Paris Olympics:
  • Individual 3m springboard - men & women
  • Individual 10m platform - men & women
  • Synchronized 3m springboard - men & women
  • Synchronized 10m platform - men & women
  • World Aquatics is the IF that governs the sport of aquatics
  • Within the sport of aquatics are multiple disciplines – artistic swimming, diving,marathon swimming, swimming, and water polo.
  • Within the discipline of diving are eight medal events:

参考资料[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/