2023年美国HIMCM中学数学建模竞赛C题

文章目录[隐藏]

中文赛题：准备奥林匹克奖牌颁奖仪式
Problem:Preparing for Olympic Medal Ceremonies

中文赛题：准备奥林匹克奖牌颁奖仪式

注意：只有所有成员年龄不满14岁半的团队才能选择C题。

下一届奥林匹克运动会将由法国巴黎主办，计划于2024年7月26日至8月11日举行。国际奥林匹克委员会（IOC）希望通过订购足够的奖牌和旗帜来为颁奖仪式做准备。IOC希望他们有足够的奖牌和旗帜供所有仪式使用，以避免出现尴尬的局面。然而，IOC也不希望因为订购过多用不到的奖牌和旗帜而浪费金钱。因此，IOC已要求你们MidMCM奥林匹克顾问团队提供模型，以便他们确定需要多少奖牌和旗帜。

为了帮助你们入手，MidMCM奥林匹克顾问的研究团队已经整理了关于巴黎2024年奥运会的以下相关信息：

参与国家: 总共已向203个有资格参赛的国家奥林匹克委员会（NOCs）发出邀请，邀请他们的国家、地区或独立运动员团体参加2024年巴黎奥林匹克运动会。因此，在2024年巴黎奥林匹克运动会上可能有多达203个代表团，每个代表团都有自己独立的旗帜^[1]。
运动项目: 2024年夏季奥林匹克运动会包括40个体育项目^[2]。每个体育项目可能包含多个比赛项目，每个比赛项目都会颁发奖牌，因此巴黎奥运会将有329个奖牌项目^[3]。这些比赛项目可能是某项运动的个人比赛和/或团队比赛，例如游泳有不同的个人比赛项目（自由泳、蝶泳等）和不同的距离（100米、200米等），以及团队接力赛。此外，许多体育项目还分男子、女子和男女混合比赛。
奖牌仪式: 每项奥林匹克体育赛事都颁发三种级别的奖牌，铜牌颁发给第三名，银牌颁发给第二名，金牌颁发给第一名。对于每场颁奖仪式，奖牌被颁发给个人运动员或团队的每位成员。此外，颁奖时领奖台上方还会升起与获奖个人或团队相对应的旗帜。
比赛场馆: 2024年巴黎奥运会将在37个比赛场馆举行奥林匹克体育项目^[4]。大多数比赛场馆位于巴黎及周边地区，但也有一些比赛场馆分布在全国各地乃至海外的塔希提岛场馆^[5]。

尽管这些研究可能为您提供了一个起点，但您的团队可能还想补充更多信息。进行自己的研究将帮助你更全面地了解情况，并开发出对IOC最有帮助的模型。

要求

1. 起始。选择以下三个场馆中的一个：La Défense Arena（游泳和水球）、Bercy Arena（体操、艺术体操和蹦床、以及篮球）；Stade de France（田径和橄榄球）。

制定你预计会在所选场馆举行的奖牌颁奖仪式日程安排。在制定日程安排时，您可能需要进行一些假设。
使用你的日程安排和其他任何有用的信息来创建一个模型，以确定您所选场馆将需要多少种类型的奖牌（金牌、银牌和铜牌）。在这样做时，您可能需要考虑以下几点：
1. 该场馆可能需要每种奖牌的最少数量是多少？
2. 该场馆可能需要每种奖牌的最大数量是多少？
3. 该场馆应该准备每种奖牌的最佳数量是多少？
使用你的日程安排和其他任何有用的信息来创建一个模型，以确定您所选场馆将需要什么旗帜，以及每种旗帜需要多少。在这样做时，您可能需要考虑以下几点：
1. 该场馆可能需要哪些旗帜？
2. 该场馆可能需要每种旗帜的最少数量是多少？
3. 该场馆可能需要每种旗帜的最大数量是多少？
4. 该场馆应该准备哪些旗帜，以及每种旗帜的最佳数量是多少？

2. 创建你的模型。将你的模型应用于上述列出的另外两个场馆。

利用你从应用奖牌模型中学到的经验，创建一个适用于所有三个场馆所需奖牌的模型。
利用你从应用旗帜模型中学到的经验，创建一个适用于所有三个场馆所需旗帜的模型。

3. 分享您的模型。写一封一到两页的信给IOC，解释你的模型如何确定2024年巴黎奥林匹克奖牌仪式所需的合适数量的奖牌和旗帜。解释你的模型如何提供一个合理的答案，以满足IOC避免尴尬和浪费金钱的需求。

4. 反思。你的模型是否可以用于2024年巴黎夏季奥林匹克运动会的所有37个比赛场馆的完整日程？你的模型是否可以供IOC用于计算他们将需要多少旗帜，以供将来的夏季奥林匹克运动会使用？您的模型是否可以供IOC用于冬季奥林匹克运动会，以便他们知道他们将需要多少旗帜用于冬季奥林匹克运动会的颁奖仪式？（你不需要在这些方面应用您的模型。您只需要考虑这样做的可能性，并简要描述您认为是否可行。请务必解释你的看法！）

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。
解决方案和要求的完整解决方案。参见原版PDF英文赛题末尾的MidMCM指南。
给IOC的一到两页信。
参考文献列表（例如，您用于收集信息的所有网站）。
AI使用报告（如果没有使用AI，则不需要）。

注意：MidMCM 提交的完整解决方案没有特定的最低页数要求。您可以使用最多 25 页的总页数来展示您所有的解决方案工作和任何您想包含的额外信息（例如：图纸、图表、计算、表格）。部分解决方案也是可以接受的。我们允许谨慎使用AI，如ChatGPT。如果你选择使用生成性 AI，就必须遵守 COMAPAI 使用政策：你必须在 PDF 解决方案文件的末尾添加一个额外的 AI 使用报告，这个报告不计入您解决方案的 25 页总页数限制。

术语表

假设：用于代替未知或不确定信息的假设或有根据的猜测。

参考资料

[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