2023年美國HIMCM中學數學建模競賽C題

3022 預計閱讀需要22分鍾

中文賽題：準備奧林匹克獎牌頒獎儀式

注意：隻有所有成員年齡不滿14歲半的團隊才能選擇C題。

下一屆奧林匹克運動會(hui) 將由法國巴黎主辦，計劃於(yu) 2024年7月26日至8月11日舉(ju) 行。國際奧林匹克委員會(hui) （IOC）希望通過訂購足夠的獎牌和旗幟來為(wei) 頒獎儀(yi) 式做準備。IOC希望他們(men) 有足夠的獎牌和旗幟供所有儀(yi) 式使用，以避免出現尷尬的局麵。然而，IOC也不希望因為(wei) 訂購過多用不到的獎牌和旗幟而浪費金錢。因此，IOC已要求你們(men) MidMCM奧林匹克顧問團隊提供模型，以便他們(men) 確定需要多少獎牌和旗幟。

為(wei) 了幫助你們(men) 入手，MidMCM奧林匹克顧問的研究團隊已經整理了關(guan) 於(yu) 巴黎2024年奧運會(hui) 的以下相關(guan) 信息：

  • 參與(yu) 國家: 總共已向203個(ge) 有資格參賽的國家奧林匹克委員會(hui) （NOCs）發出邀請，邀請他們(men) 的國家、地區或獨立運動員團體(ti) 參加2024年巴黎奧林匹克運動會(hui) 。因此，在2024年巴黎奧林匹克運動會(hui) 上可能有多達203個(ge) 代表團，每個(ge) 代表團都有自己獨立的旗幟[1]。

  • 運動項目: 2024年夏季奧林匹克運動會(hui) 包括40個(ge) 體(ti) 育項目[2]。每個(ge) 體(ti) 育項目可能包含多個(ge) 比賽項目，每個(ge) 比賽項目都會(hui) 頒發獎牌，因此巴黎奧運會(hui) 將有329個(ge) 獎牌項目[3]。這些比賽項目可能是某項運動的個(ge) 人比賽和/或團隊比賽，例如遊泳有不同的個(ge) 人比賽項目（自由泳、蝶泳等）和不同的距離（100米、200米等），以及團隊接力賽。此外，許多體(ti) 育項目還分男子、女子和男女混合比賽。

  • 獎牌儀(yi) 式: 每項奧林匹克體(ti) 育賽事都頒發三種級別的獎牌，銅牌頒發給第三名，銀牌頒發給第二名，金牌頒發給第一名。對於(yu) 每場頒獎儀(yi) 式，獎牌被頒發給個(ge) 人運動員或團隊的每位成員。此外，頒獎時領獎台上方還會(hui) 升起與(yu) 獲獎個(ge) 人或團隊相對應的旗幟。

  • 比賽場館: 2024年巴黎奧運會(hui) 將在37個(ge) 比賽場館舉(ju) 行奧林匹克體(ti) 育項目[4]。大多數比賽場館位於(yu) 巴黎及周邊地區，但也有一些比賽場館分布在全國各地乃至海外的塔希提島場館[5]。

盡管這些研究可能為(wei) 您提供了一個(ge) 起點，但您的團隊可能還想補充更多信息。進行自己的研究將幫助你更全麵地了解情況，並開發出對IOC最有幫助的模型。

要求

1. 起始。選擇以下三個(ge) 場館中的一個(ge) ：La Défense Arena（遊泳和水球）、Bercy Arena（體(ti) 操、藝術體(ti) 操和蹦床、以及籃球）；Stade de France（田徑和橄欖球）。

  • 製定你預計會(hui) 在所選場館舉(ju) 行的獎牌頒獎儀(yi) 式日程安排。在製定日程安排時，您可能需要進行一些假設。

  • 使用你的日程安排和其他任何有用的信息來創建一個(ge) 模型，以確定您所選場館將需要多少種類型的獎牌（金牌、銀牌和銅牌）。在這樣做時，您可能需要考慮以下幾點：

    1. 該場館可能需要每種獎牌的最少數量是多少？
    2. 該場館可能需要每種獎牌的最大數量是多少？
    3. 該場館應該準備每種獎牌的最佳數量是多少？

  • 使用你的日程安排和其他任何有用的信息來創建一個(ge) 模型，以確定您所選場館將需要什麽(me) 旗幟，以及每種旗幟需要多少。在這樣做時，您可能需要考慮以下幾點：

    1. 該場館可能需要哪些旗幟？
    2. 該場館可能需要每種旗幟的最少數量是多少？
    3. 該場館可能需要每種旗幟的最大數量是多少？
    4. 該場館應該準備哪些旗幟，以及每種旗幟的最佳數量是多少？

2. 創建你的模型。將你的模型應用於(yu) 上述列出的另外兩(liang) 個(ge) 場館。

  • 利用你從應用獎牌模型中學到的經驗，創建一個適用於所有三個場館所需獎牌的模型。
  • 利用你從應用旗幟模型中學到的經驗，創建一個適用於所有三個場館所需旗幟的模型。

3. 分享您的模型。寫(xie) 一封一到兩(liang) 頁的信給IOC，解釋你的模型如何確定2024年巴黎奧林匹克獎牌儀(yi) 式所需的合適數量的獎牌和旗幟。解釋你的模型如何提供一個(ge) 合理的答案，以滿足IOC避免尷尬和浪費金錢的需求。

4. 反思。你的模型是否可以用於(yu) 2024年巴黎夏季奧林匹克運動會(hui) 的所有37個(ge) 比賽場館的完整日程？你的模型是否可以供IOC用於(yu) 計算他們(men) 將需要多少旗幟，以供將來的夏季奧林匹克運動會(hui) 使用？您的模型是否可以供IOC用於(yu) 冬季奧林匹克運動會(hui) ，以便他們(men) 知道他們(men) 將需要多少旗幟用於(yu) 冬季奧林匹克運動會(hui) 的頒獎儀(yi) 式？（你不需要在這些方麵應用您的模型。您隻需要考慮這樣做的可能性，並簡要描述您認為(wei) 是否可行。請務必解釋你的看法！）

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。
  • 解決方案和要求的完整解決方案。參見原版PDF英文賽題末尾的MidMCM指南。
  • 給IOC的一到兩頁信。
  • 參考文獻列表（例如，您用於收集信息的所有網站）。
  • AI使用報告（如果沒有使用AI，則不需要）。

注意：MidMCM 提交的完整解決(jue) 方案沒有特定的最低頁數要求。您可以使用最多 25 頁的總頁數來展示您所有的解決(jue) 方案工作和任何您想包含的額外信息（例如：圖紙、圖表、計算、表格）。部分解決(jue) 方案也是可以接受的。我們(men) 允許謹慎使用AI，如ChatGPT。如果你選擇使用生成性 AI，就必須遵守 COMAP AI 使用政策：你必須在 PDF 解決(jue) 方案文件的末尾添加一個(ge) 額外的 AI 使用報告，這個(ge) 報告不計入您解決(jue) 方案的 25 頁總頁數限製。

術語表

假設：用於(yu) 代替未知或不確定信息的假設或有根據的猜測。

參考資料

[1]One Year to Go: IOC invites NOCs and their best athletes to the Olympic Games Paris 2024:https://olympics.com/ioc/news/one-year-to-go-ioc-invites-nocs-and-their-best-athletes-to-the-olympic-games-paris-2024

[2]Sports:https://olympics.com/en/sports/summer-olympics#paris-2024

[3]Olympic Games Paris 2024: Full schedule and day-by-day competitions:https://olympics.com/en/news/olympic-games-paris-2024-full-schedule-and-day-by-day-competitions

[4]Paris 2024 Olympics venue map:https://www.parisdigest.com/sports/paris-olympics-2024.htm#:~:text=Paris%20Olympics%20venue%20map%20locates,26%20to%20August%2011%202024

[5]Competition venue concept:https://www.paris2024.org/en/competition-venue-concept

Problem:Preparing for Olympic Medal Ceremonies

Note: Only teams with all members younger than 14 ½ years old may choose Problem C.

The next Olympic Games are being hosted by Paris, France and will be held 26 July - 11 August 2024. The International Olympic Committee (IOC) wants to prepare by ordering enough medals and flags for the medal ceremonies. The IOC wants to make sure they have enough medals and flags for all the ceremonies to avoid being embarrassed. The IOC also does not want to waste money, however, by ordering too many medals and flags that will not be used. Therefore, the IOC has asked your team at MidMCM Olympic Consultants to provide models they can use to determine how many medals and flags they will need.

The research team at MidMCM Olympic Consultants has compiled the following information about the Paris 2024 Olympics to help you get started.

  • Participating Countries: In total, invitations to the Paris 2024 Olympic Games have been sent to 203 eligible National Olympic Committees (NOCs) inviting their countries, regions, territories, or unaffiliated groups of athletes to compete. Therefore, there may be up to 203 delegations of athletes, each represented by their own unique flag, at the Paris 2024 Olympics.[1]

  • Olympic Sports: The summer Olympics for 2024 consists of 40 sports.[2]Sports may consist of multiple events that each receive medals, so the Paris Olympics will have 329 medal events.[3]The events may be individual and/or team events for a particular sport. Swimming, for example, has different individual events for each of the strokes (freestyle, butterfly, etc.) and different distances (100m, 200m, etc.), as well as relay events where a team of swimmers participate. Additionally, many sports have men’s, women’s, and mixed gender versions of the events.

  • Medal Ceremonies: Each Olympic sporting event awards three levels of medals, Bronze for 3rd place, Silver for 2nd place, and Gold for 1st place. For each medal ceremony, medals are awarded to the individual athletes or to each individual member of a team. Additionally, the flag under which an individual or team is competing is flown above the podium stand for each individual or team winning any level medal.

  • Venues: The Paris 2024 Olympics will be hosted at 37 venues where the Olympic sports will take place.[4]Most of the venues are centered in and around Paris, but there are also venues spread across the country and even overseas to Tahiti.[5]

While this research may get you started, your team will likely want to add to it. Doing your own research will help you to understand the situation more fully and to develop models that will be most helpful for the IOC.

Requirements

1. Get Started. Choose ONE of the following three venues: i. La Défense Arena (Swimming and Water Polo); ii. Bercy Arena (Gymnastics – Artistic and Trampoline – and Basketball); iii. Stade de France (Athletics and Rugby)

  • Develop a schedule of the medal ceremonies you would expect to happen at the venue you selected. In making your schedule you will likely need to make some assumptions.

  • Use your schedule and any other helpful information to create a model to determine how many of each type of medal (gold, silver, and bronze) your selected venue will need. In doing so, you may want to consider:

    1. What is the minimum number of each kind of medal that venue could need?
    2. What is the maximum number of each kind of medal that venue could need?
    3. What is the best number of each kind of medal that venue should have?

  • Use your schedule and any other helpful information to create a model to determine what flags – and how many of each – your selected venue will need. In doing so, you may want to consider:

    1. What flags could your selected venue possibly need?
    2. What is the minimum number of each kind of flag that venue could need?
    3. What is the maximum number of each kind of flag that venue could need?
    4. What flags should your selected venue have and what is the best number of each of those kinds of flags?

2. Create your Model. Apply your models to the other two venues listed above.

  • Use what you learn from applying your medal model to create a model for the medals needed for all three venues.
  • Use what you learn from applying your flag model to create a model for the flags needed for all three venues.

3. Share your Model. Create a one- to two-page letter to the IOC explaining how your model can determine an appropriate number of medals and flags for the Paris 2024 Olympic medal ceremonies. Explain how your model provides a reasonable answer that satisfies the IOC’s need to avoid embarrassment and avoid wasting money.

4.Reflect. Can your model be used for the full schedule of Paris Summer 2024 Olympics at all 37 venues? Can your model be used by the IOC to know how many flags they will need for future Summer Olympics? Can your model be used by the IOC for the Winter Olympics so that they know how many flags they will need for Winter Olympics medal ceremonies? (Youdo notneed to apply your model in any of these ways. Instead, you should think about the possibility of doing so and briefly describe whether you think it will work. Be sure to explain why you think what you do!)

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

  • One-page Summary Sheet.
  • Table of Contents.
  • Your complete solution to the problem and requirements. See MidMCM Guidance at the end of this document.
  • One- to two-page letter to the IOC.
  • References list (for example, any websites you used to gather information).
  • AI Use Report (If used)

Note: There is no specific required minimum page length for a complete MidMCM 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

Assumptions: hypotheses or educated guesses that take the place of an unknown or uncertain piece of information.

References

[1]One Year to Go: IOC invites NOCs and their best athletes to the Olympic Games Paris 2024:https://olympics.com/ioc/news/one-year-to-go-ioc-invites-nocs-and-their-best-athletes-to-the-olympic-games-paris-2024

[2]Sports:https://olympics.com/en/sports/summer-olympics#paris-2024

[3]Olympic Games Paris 2024: Full schedule and day-by-day competitions:https://olympics.com/en/news/olympic-games-paris-2024-full-schedule-and-day-by-day-competitions

[4]Paris 2024 Olympics venue map:https://www.parisdigest.com/sports/paris-olympics-2024.htm#:~:text=Paris%20Olympics%20venue%20map%20locates,26%20to%20August%2011%202024

[5]Competition venue concept:https://www.paris2024.org/en/competition-venue-concept

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