2025年美國大學生數學建模競賽C題中英版

中文賽題：奧運獎牌榜模型

在 2024 年巴黎舉(ju) 行的最新一屆夏季奧運會(hui) 上，除了觀看個(ge) 人項目外，粉絲(si) 們(men) 還關(guan) 注了每個(ge) 國家的整體(ti) “獎牌榜”。最終結果（表 1）顯示，美國的總獎牌數最多（126 枚），中國和美國並列第一金牌榜首位（40 枚）。東(dong) 道主法國在金牌數（16 枚）排名中位居第五，但在獎牌總數方麵排名第四，而英國以 14 枚金牌排名第七，在獎牌總數方麵排名第三。

排行榜頂端的排名總是受到密切關(guan) 注，但其他國家的獎牌數往往也同樣受到重視。例如，阿爾巴尼亞(ya) （2 枚獎牌）、佛得角、多米尼加和聖盧西亞(ya) （2 枚獎牌）在巴黎奧運會(hui) 上贏得了本國首枚奧運獎牌。多米尼加和聖盧西亞(ya) 也分別在這屆奧運會(hui) 上獲得了一枚金牌。仍有 60 多個(ge) 國家尚未贏得奧運獎牌。人們(men) 通常會(hui) 預測最終的獎牌數，但通常不是基於(yu) 曆史獎牌數，而是在即將舉(ju) 行的奧運會(hui) 開始前，屆時將公布當前計劃參賽的運動員（例如：https://www.nielsen.com/news-center/2024/virtual-medal-table-forecast/ ）。

提供了所有夏季奧運會(hui) 獎牌榜、主辦國以及所有夏季奧運會(hui) 各屆奧運會(hui) 按運動項目細分的奧運賽事數量數據。此外，還提供了所有個(ge) 人奧運會(hui) 參賽者及其賽事和結果（獎牌類型或無獎牌）的數據。您的模型和數據分析必須僅(jin) 使用提供的數據集。您可以使用其他資源來提供背景和上下文或幫助解釋結果（請務必記錄來源）。具體(ti) 來說，使用提供的數據來：

• 為每個國家（至少為金牌和獎牌總數）開發一個獎牌計數模型。包括模型預測的不確定性/精確度的估計值和模型表現的衡量標準。
  • 根據您的模型，您對 2028 年美國洛杉磯夏季奧運會獎牌榜的預測是什麽？包括所有結果的預測區間。您認為哪些國家最有可能進步？哪些國家的表現會比 2024 年更差？
  • 您的模型應包括尚未獲得獎牌的國家；您預測有多少國家將在下屆奧運會上獲得第一枚獎牌？您對這個估計的幾率是多少？
  • 您的模型還應考慮特定奧運會的賽事（數量和類型）。探索賽事與國家獲得的獎牌數之間的關係。哪些運動對各個國家最重要？為什麽？本國選擇的項目如何影響結果？
• 運動員可以代表不同的國家參賽，但由於公民身份要求，他們很難更換國家。然而，教練可以輕鬆地從一個國家轉移到另一個國家，因為他們不需要是公民就可以執教。因此，有可能出現“偉大教練”效應。兩個可能的例子包括郎平^[2]，他曾執教美國和中國的排球隊，並奪得冠軍；以及有時備受爭議的體操教練貝拉·卡羅利^[3]，他先後執教羅馬尼亞和美國女子隊，取得了巨大的成功。檢查數據，尋找可能因“偉大教練”效應而發生變化的證據。你估計這種影響對獎牌數的貢獻有多大？選擇三個國家，確定哪些運動項目應該考慮投資“偉大”教練，並估計這種影響。
• 您的模型還揭示了有關奧運獎牌數量的哪些其他原創見解？解釋這些見解如何為國家奧委會提供信息。

您的 PDF 解決(jue) 方案總頁數不得超過25頁，包括：

• 一頁摘要頁。
• 目錄。
• 完整解決方案。
• 一封給交通官員的信（一頁）。
• 參考文獻列表。
• AI使用報告（如果使用不計入 25 頁總限製）。

注意：完整的 MCM 提交文件沒有特定的最低頁數要求。您最多可使用 25 頁來包含您的所有解決(jue) 方案內(nei) 容及其他補充信息（例如圖紙、圖表、計算和表格）。允許提交部分解決(jue) 方案。我們(men) 允許謹慎使用 AI 工具（例如 ChatGPT），但完成本問題並不一定需要使用 AI 工具。如果您選擇使用生成式 AI 工具，必須遵循 COMAP 的 AI使用政策。此政策要求您在解決(jue) 方案文件的最後額外添加一份 AI 使用報告，不計入 25 頁的總限製。

全新 MCM/ICM：在線提交流程

本文旨在協助和指導參加 MCM/ICM 的學生和顧問。在文章中，COMAP 提供了有關(guan) 使用新在線提交頁麵的新在線提交流程的信息：

https://forms.comap.org/241335097294056

您需要團隊的控製號、顧問 ID 號和問題選擇才能完成提交。

數據文件

2025_Problem_C_Data.zip：此 zip 文件包含下麵列出的所有 5 個(ge) 數據文件。

• data_dictionary.csv – 帶有示例的數據庫描述
• summerOly_athletes.csv – 所有參賽者及其運動項目、年份和結果（獎牌類型或無）
• summerOly_medal_counts.csv – 1896 年至 2024 年所有夏季奧運會的完整國家獎牌統計表
• summerOly_hosts.csv – 1896 年至 2032 年所有夏季奧運會主辦國名單
• summerOly_programs.csv – 1896 年至 2032 年所有夏季奧運會按運動項目/學科統計的賽事數量和總數

國家名稱等數據由國際奧林匹克委員會(hui) (IOC)（在其 Olympics.com 網站上）在特定奧運會(hui) 期間記錄。因此，數據集中的名稱可能會(hui) 發生變化。與(yu) 所有數據一樣，記錄可能存在異常。例如，請注意，在運動員數據集中，在某些情況下，對於(yu) 網球、乒乓球、沙灘排球等運動，“球隊”包含比國家更多的細節。例如，德國-1 將是 2000 年奧運會(hui) 上來自德國的兩(liang) 支沙灘排球隊中的第一支。關(guan) 於(yu) 如何處理數據的決(jue) 策和假設是建模過程的重要組成部分。

術語表

國際奧林匹克委員會(hui) (IOC)：是管理奧運會(hui) 和奧林匹克運動的國際非政府體(ti) 育機構。IOC 最為(wei) 人熟知的是負責組織夏季和冬季奧運會(hui) 的組織。

奧運會(hui) 項目：是 IOC 為(wei) 每屆奧運會(hui) 製定的所有體(ti) 育比賽項目。

SDE：運動、分項或賽事

• 運動項目：國際奧委會將奧林匹克運動項目定義為由單一國際體育聯合會 (IF) 管理的項目。一項運動項目可能包含一個或多個項目，每個項目都是一個或多個賽事的焦點。
• 學科：包括一項或多項賽事的運動分支。
• 賽事：學科內的比賽，可產生排名和獎項（例如獎牌）。
• 2024 年巴黎奧運會項目中運動、學科和賽事之間關係的示例：
  • 個人 3 米跳板 - 男子和女子
  • 個人 10 米跳台 - 男子和女子
  • 雙人 3 米跳板 - 男子和女子
  • 雙人 10 米跳台 - 男子和女子
  • 世界水上運動聯合會是管理水上運動的國際體育聯合會
  • 水上運動包括多個項目——花樣遊泳、跳水、馬拉鬆遊泳、遊泳和水球。
  • 跳水項目包括八個獎牌項目：

免責聲明

COMAP 是一家致力於(yu) 改善數學教育的非盈利組織，重點是提高學生的數學建模能力。本次競賽問題參考了國際奧林匹克委員會(hui) (IOC) 的材料。我們(men) 承認並尊重 IOC 對這些材料的所有權，此處僅(jin) 將其用於(yu) 教育、非商業(ye) 目的，以豐(feng) 富參與(yu) 者的學習(xi) 體(ti) 驗。本內(nei) 容不受 IOC 認可或隸屬於(yu) 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:

參考資料[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/