2025 MCM
Problem A: Testing Time: The Constant Wear On Stairs 2025 年 MCM
问题 A：测试时间：楼梯的持续磨损
Stone is a symbol of steadfast permanence, and carved rock is used as a building material because of its ability to resist wear. Despite its durability, even stone is not impervious to wear. One of the few things more resilient is the persistence of people. 石头是坚定不移的永恒象征，而雕刻的岩石因其抗磨损能力而被用作建筑材料。尽管它很耐用，但即使是石头也不是不透磨损的。为数不多的更有弹性的因素之一是人们的坚持。
Figure 1: Example of steps that have uneven wear after long term use. 图 1：长期使用后磨损不均匀的步骤示例。
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 particularsite 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. 用于制作台阶的石头和其他材料会受到持续的长期磨损，并且磨损可能不均匀。例如，非常古老的寺庙和教堂的楼梯可能比其边缘磨损得更多，并且踏板不再有水平的顶部，但看起来是弯曲的。由于这种结构的性质，这些建筑物往往在很长一段时间内都有人居住，但特定地点的人的存在往往早于建筑物的建造。这使得难以准确确定结构的建造日期。当结构的施工时间很长、进行了翻新并且随着时间的推移向结构添加了新部件时，情况会变得更加复杂。
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. 您的团队需要提供指导，说明考古学家可以从一组破旧的楼梯中确定哪些信息。楼梯可以由各种材料制成，例如石头或木材。此外，可能会有一组楼梯建造时间的估计值，但是，可能很难获得准确的时间。从历史记录中可能不清楚每个时期建造了结构中的哪组楼梯。
In addition to age, an archaeologist may also be interested in determining the traffic patternsassociated 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? 除了年龄之外，考古学家可能还对确定与楼梯间使用方式相关的交通模式感兴趣。例如，人们是否同时上下楼梯，还是在任何给定时间使用主要方向？考古学家可能还想确定楼梯井的使用频率。例如，是很多人在短时间内使用它，还是少数人在很长一段时间内使用它？
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: 您的团队需要开发一个模型，该模型可以确定给定一组楼梯可以得出哪些基本结论。根据一组特定楼梯的磨损模式，您的模型应提供一些基本预测：
How often were the stairs used? 楼梯多久使用一次？
Was a certain direction of travel favored by the people using the stairs? 使用楼梯的人是否喜欢某个特定的前进方向？
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?) 有多少人同时使用楼梯？（例如，两人并排爬楼梯还是单列行走？
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 is non-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. 您可以假设考古学家可以访问相关结构，并且可以获得您的团队认为重要的任何测量值。测量必须以非破坏性的方式进行，成本必须相对较低，并且测量可以由一小群人使用最少的工具进行。您应该明确需要哪些测量。
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: 还有其他问题可能更难解决。假设存在对年龄的估计值、楼梯间的使用方式以及对结构中日常生活模式的估计值，请确定可以为以下问题提供哪些指导：
Is the wear consistent with the information available? 磨损是否与可用信息一致？
What is the age of the stairwell and how reliable is the estimate? 楼梯间的使用年限是多少，估计的可靠性如何？
What repairs or renovations have been conducted? 进行了哪些维修或翻新？
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? 能否确定材料的来源？例如，如果使用了石头，磨损是否与考古学家认为是原始来源的采石场的材料一致，或者如果使用了木材，磨损是否与假定使用的树木的年龄和类型一致？
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? 关于典型一天中使用楼梯的人数，可以确定哪些信息，以及在短时间内使用楼梯的人数是大量，还是在较长时间内使用少数人？
Your PDF solution of no more than 25 total pages should include: 总页数不超过 25 页的 PDF 解决方案应包括：
One-page Summary Sheet. 单页摘要表。
Table of Contents. 目录。
Your complete solution. 您的完整解决方案。
References list. 引用列表。
AI Use Report (If used does not count toward the 25-page limit.) AI 使用报告（如果使用，则不计入 25 页的限制。
Note: There is no specificrequired 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. 注意： 完整 MCM 提交没有具体要求的最短页面长度要求。您最多可以使用总共 25 页来存储所有解决方案工作和要包含的任何其他信息（例如：图纸、图表、计算、表格）。接受部分解决方案。我们允许谨慎使用 ChatGPT 等 AI，尽管没有必要为此问题创建解决方案。如果您选择使用生成式 AI，则必须遵循 COMAP AI 使用政策。这将产生一份额外的 AI 使用报告，您必须将其添加到 PDF 解决方案文件的末尾，并且不会计入解决方案的总页数限制为 25 页。
NEW MCM/ICM: Online Submission Process 新的 MCM/ICM：在线提交流程
The purpose of this article is to assist and guide students and advisors participating in HiMCM/MidMCM. In the article, COMAP, provides information about the new online submission process using the new online submission page 本文的目的是帮助和指导参与 HiMCM/MidMCM 的学生和顾问。在文章 COMAP 中，提供了有关使用新的在线提交页面的新在线提交流程的信息 https://forms.comap.org/241335097294056. You will need your team’s control number, advisor id number and your problem choice to complete your submission. https://forms.comap.org/241335097294056。您需要团队的控制编号、顾问 ID 编号和您的问题选择才能完成提交。
Glossary 词汇表
Permanence: The state or quality of lasting or remaining unchanged indefinitely. 持久性：持久或无限期保持不变的状态或质量。
Impervious: Unable to be affected by. Impervious：无法受到影响。
Non-destructive: Not involving damage or destruction, especially of an object or material that is being evaluated. Non-destructive（非破坏性）：不涉及损坏或破坏，尤其是对正在评估的对象或材料。
