2025年美國大學生數學建模競賽A題中英版

中文賽題：測試時間：樓梯的持續磨損

石頭象征著堅定的永恒，由於(yu) 其抵抗磨損的能力，它被用作建築材料。然而，即使是石頭也無法完全免受磨損。唯一比石頭更能經受考驗的，是人類的堅持不懈。

樓梯所用的石頭及其他材料長期以來不斷受到磨損，這種磨損可能會(hui) 出現不均勻的情況。例如，在一些極為(wei) 古老的寺廟和教堂中，樓梯中央部分的磨損可能比邊緣部分更嚴(yan) 重，使踏板頂部不再平整，而可能呈現彎曲的形態。由於(yu) 這些建築物的性質，它們(men) 往往在很長一段時間內(nei) 有人居住，但人們(men) 出現在該地點的時間往往早於(yu) 建築物的建造時間。這使得準確判斷某一建築物的建造日期變得非常困難。

您的團隊被要求提供指導，以幫助考古學家從(cong) 一組磨損的樓梯中得出信息。這些樓梯可能由多種材料建造，例如石材或木材。關(guan) 於(yu) 樓梯的建造時間可能有一個(ge) 估計值，但很難得出一個(ge) 精確的時間點。曆史記錄可能並未清晰表明某一時間段內(nei) 建築物中的哪組樓梯被建造。

除了年代之外，考古學家可能還對確定與(yu) 樓梯間使用方式相關(guan) 的交通模式感興(xing) 趣。例如，人們(men) 是否同時上下樓梯，還是在任何給定時間都使用一個(ge) 主要方向？考古學家可能還想確定樓梯間的使用頻率。例如，是否有很多人在短時間內(nei) 使用它，還是隻有少數人在很長一段時間內(nei) 使用它？

您的團隊被要求開發一個(ge) 模型，用於(yu) 根據某組樓梯的磨損模式得出一些基本結論：

• 樓梯的使用頻率是多少？
• 使用樓梯時是否存在某一方向的偏好？
• 同一時間有多少人使用樓梯？（例如，兩人並排行走或單行通過？）

您可以假設考古學家可以接觸到相關(guan) 結構，並可以獲得您的團隊認為(wei) 重要的任何測量結果。測量必須以非破壞性的方式進行，成本必須相對較低，並且可以由一小隊人使用最少的工具進行測量。您應該明確需要進行哪些測量。

還有一些更複雜的問題需要解決(jue) 。假設有樓梯使用方式的相關(guan) 信息、樓梯年齡的估計值，以及關(guan) 於(yu) 建築日常生活模式的信息，請回答以下問題：

• 磨損是否與現有信息一致？
• 樓梯的年齡估算值是多少？估算的可靠性如何？
• 是否可以判斷樓梯經曆了哪些維修或翻新？
• 是否可以確定材料的來源？例如，石材的磨損是否與考古學家認為的采石場材料一致？如果使用的是木材，其磨損是否與樹種和樹齡的假設相符？
• 是否可以推斷典型一天中使用樓梯的人數？是有大量人在短時間內使用過樓梯，還是少量人在較長時間內使用過樓梯？

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。

術語表

永恒：無限期地持續或保持不變的狀態或性質。

免受：無法受到影響。

非破壞性：不涉及損壞或破壞，尤其是對正在評估的物體(ti) 或材料。

Problem: Testing Time: The Constant Wear On Stairs

Stone is a symbol of steadfastpermanence, and carved rock is used as a building material because of its ability to resist wear. Despite its durability, even stone is notimperviousto wear. One of the few things more resilient is the persistence of people.

The stone and other materials used to create steps are subject to constant, long-term wear, and the wear can be uneven. For example, extremely old temples and churches may have stairs in which the center of the steps has been worn down more than their edges, and the treads no longer have a level top but can appear to be bowed. Due to the nature of such structures, the buildings tend to have been inhabited by people over a very long period, but the presence of people at a particular site often predates the construction of buildings. This can make it difficult to accurately determine the date a structure was constructed. The situation is further complicated when construction on a structure occurs over a very long time, renovations have been made, and new parts to the structure have been added over time.

Your team is asked to provide guidance on what information archaeologists can determine from a set of worn stairs. The stairs may be constructed from various materials, for example stone or wood. Also, an estimate of when a set of stairs was constructed may be available, however, it may be difficult to get a precise time. It may not be clear from the historical records which set of stairs in a structure was constructed in each time period.

In addition to age, an archaeologist may also be interested in determining the traffic patterns associated with how a stairwell was used. For example, were people moving up as well as down the stairs at the same time or was a predominant direction used at any given time? An archaeologist may also want to determine how often the stair well was used. For example, did a large number of people use it over a short period or did a few people use it over a very long period?

Your team is asked to develop a model that can determine what basic conclusions can be drawn given a set of stairs. Your model should provide some basic predictions given the patterns of wear on a particular set of stairs:

• How often were the stairs used?
• Was a certain direction of travel favored by the people using the stairs?
• How many people used the stairs simultaneously? (For example, did pairs of people climb the stairs side-by-side or did they travel single file?)

You can assume that an archaeologist has access to the structure in question and can obtain whatever measurement your team believes is important. The measurements must be made in a way that isnon-destructive, the cost must be relatively low, and the measurements can be taken by a small team of people with minimal tools. You should make clear what measurements are required.

There are other questions that may be more difficult to address. Assuming an estimate of the age exists, the way the stairwell was used, and an estimate on the daily patterns of life in the structure are available, determine what guidance can be provided to the following questions:

• Is the wear consistent with the information available?
• What is the age of the stairwell and how reliable is the estimate?
• What repairs or renovations have been conducted?
• Can the source of the material be determined? For example, if stone is used is the wear consistent with materials from a quarry the archaeologist believes is the original source or if wood was used is the wear consistent with the age and type of trees that are assumed to be used?
• What information can be determined with respect to the numbers of people using the stairs in a typical day and were there large numbers of people using the stairs over a short time or a small number of people over a longer time?

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.

Glossary

Permanence: The state or quality of lasting or remaining unchanged indefinitely.

Impervious: Unable to be affected by.

Non-destructive: Not involving damage or destruction, especially of an object or material that is being evaluated.