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1、120122012 年美国国际大学生数年美国国际大学生数 学建模竞赛学建模竞赛(MCM(MCM ICM)ICM)题目题目 翻译翻译IMPORTANT CHANGE TO CONTEST RULES FOR MCM/ICM 2012: Teams (Student or Advisor) are now required to submit an electronic copy (summary sheet and solution) of their solution paper by email to . Your email MUST be received at COMAP by the
2、 submission deadline of 8:00 PM EST, February 13, 2012. Teams are free to choose between MCM Problem A, MCM Problem B or ICM Problem C. COMAP Mirror Site: For more in: http:/ MCM: The Mathematical Contest in Modeling ICM: The Interdisciplinary Contest in Modeling 2012 Contest Problems 2MCM PROBLEMS
3、PROBLEM A: The Leaves of a Tree much do the leaves on a tree weigh?How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following: ?
4、Why do leaves have the various shapes that they have? ? Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the “volume” of the tree and its branches effect the shape? ? Speaking of profiles, is leaf shape (general
5、 characteristics) related to tree profile/branching structure? ? How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)? In addition to your one page summary sheet prepare a o
6、ne page letter to an editor of a scientific journal outlining your key findings. 2012 美赛美赛 A 题：一棵树的叶子题：一棵树的叶子 （数学中国（数学中国3翻译）翻译） “一棵树的叶子有多重？一棵树的叶子有多重？”怎么能估计树的叶子怎么能估计树的叶子（或者树的任何其它部分）的实际重量？怎样对叶子进行（或者树的任何其它部分）的实际重量？怎样对叶子进行分类？建立一个数学模型来对叶子进行描述和分类。模型分类？建立一个数学模型来对叶子进行描述和分类。模型要考虑和回答下面的问题：要考虑和回答下面的问题： ? 为什么叶子
7、具有各种形状？为什么叶子具有各种形状？? 叶子之间是要将相互重叠的部分最小化，以便可以最叶子之间是要将相互重叠的部分最小化，以便可以最大限度的接触到阳光吗？树叶的分布以及树干和枝杈的体大限度的接触到阳光吗？树叶的分布以及树干和枝杈的体积影响叶子的形状吗？积影响叶子的形状吗？ 就轮廓来讲，叶形（一般特征）是和树的轮廓以及分枝就轮廓来讲，叶形（一般特征）是和树的轮廓以及分枝结构有关吗？结构有关吗？ ? 你将如何估计一棵树的叶子质量？叶子你将如何估计一棵树的叶子质量？叶子的质量和树的尺寸特征（包括和外形轮廓有关的高度、质的质量和树的尺寸特征（包括和外形轮廓有关的高度、质量、体积）有联系吗？量、体积）
8、有联系吗？ 除了你的一页摘要以外，给科学杂志的编辑写一封信，除了你的一页摘要以外，给科学杂志的编辑写一封信，阐述你的主要发现。阐述你的主要发现。 PROBLEM B: Camping along the Big Long River Visitors to the Big Long River (225 miles) can enjoy scenic views and exciting white water rapids. The river is inaccessible to hikers, so the only way to enjoy it is to take a river
9、trip that requires several days of camping. River trips all start at First Launch 4and exit the river at Final Exit, 225 miles downstream. Passengers take either oar- powered rubber rafts, which travel on average 4 mph or motorized boats, which travel on average 8 mph. The trips range from 6 to 18 n
10、ights of camping on the river, start to finish The government agency responsible for managing this river wants every trip to enjoy a wilderness experience, with minimal contact with other groups of boats on the river. Currently, X trips travel down the Big Long River each year during a six month per
11、iod (the rest of the year it is too cold for river trips). There are Y camp sites on the Big Long River, distributed fairly uniformly throughout the river corridor. Given the rise in popularity of river rafting, the park managers have been asked to allow more trips to travel down the river. They wan
12、t to determine how they might schedule an optimal mix of trips, of varying duration (measured in nights on the river) and propulsion (motor or oar) that will utilize the campsites in the best way possible. In other words, how many more boat trips could be added to the Big Long Rivers rafting season?
13、 The river managers have hired you to advise them on ways in which to develop the best schedule and on ways in which to determine the carrying capacity of 5the river, remembering that no two sets of campers can occupy the same site at the same time. In addition to your one page summary sheet, prepar
14、e a one page memo to the managers of the river describing your key findings. 问题问题B：沿长江大露营：沿长江大露营 到到 Big Long River（225 英里）游玩的游客可以享受那里英里）游玩的游客可以享受那里的风景和振奋人心的急流。远足者没法到达这条河，唯一的风景和振奋人心的急流。远足者没法到达这条河，唯一去的办法是漂流过去。这需要几天的露营。河流旅行始于去的办法是漂流过去。这需要几天的露营。河流旅行始于First Launch，在，在 Final Exit 结束，共结束，共 225 英里的顺流。旅英里的顺流
15、。旅客可以选择依靠船桨来前进的橡皮筏，它的速度是客可以选择依靠船桨来前进的橡皮筏，它的速度是 4 英里英里每小时，或者选择每小时，或者选择 8 英里每小时的摩托船。旅行从开始到英里每小时的摩托船。旅行从开始到结束包括大约结束包括大约 6 到到 18 个晚上的河中的露营。负责管理这条个晚上的河中的露营。负责管理这条河的政府部门希望让每次旅行都能尽情享受野外经历，同河的政府部门希望让每次旅行都能尽情享受野外经历，同时能尽量少的与河中其他的船只相遇。当前，每年经过时能尽量少的与河中其他的船只相遇。当前，每年经过 Big Long 河的游客有河的游客有 X 组，这些漂流都在一个以组，这些漂流都在一个以
16、 6 个月长短的个月长短的时期内进行，一年中的其他月份非常冷，不会有漂流。在时期内进行，一年中的其他月份非常冷，不会有漂流。在Big Long 上有上有 Y 处露营地点，平均分布于河廊。随着漂流处露营地点，平均分布于河廊。随着漂流人数的增加，管理者被要求应该允许让更多的船只漂流。人数的增加，管理者被要求应该允许让更多的船只漂流。他们要决定如何来安排最优的方案：包括旅行时间（以在他们要决定如何来安排最优的方案：包括旅行时间（以在河上的夜晚数计算）河上的夜晚数计算） 、选择哪种船（摩托还是桨船）、选择哪种船（摩托还是桨船） ，从而，从而能够最好地利用河中的露营地。换句话说，能够最好地利用河中的露营地。换句话说，Big Long River在漂流季节还能增加多少漂流旅行数？管理者希望你能给在漂流季节还能增加多少漂流旅行数？管理者希望你能给6他们最好的建议，告诉他们如何决定河流的容纳量，记住他们最好的建议，告
