美赛历年试题.docx

1、美赛历年试题2009 年美国大学生数学建模竞赛 MCM、ICM 试题2009 MCM A: Designing a Traffic CircleMany cities and communities have traffic circlesfrom large ones with many lanes in the circle (such as at the Arc de Triomphe in Paris and the Victory Monument in Bangkok) to small ones with one or two lanes in the circle. Some

2、of these traffic circles position a stop sign or a yield sign on every incoming road that gives priority to traffic already in the circle; some position a yield sign in the circle at each incoming road to give priority to incoming traffic; and some position a traffic light on each incoming road (wit

3、h no right turn allowed on a red light). Other designs may also be possible. The goal of this problem is to use a model to determine how best to control traffic flow in, around, and out of a circle. State clearly the objective(s) you use in your model for making the optimal choice as well as the fac

4、tors that affect this choice. Include a Technical Summary of not more than two double-spaced pages that explains to a Traffic Engineer how to use your model to help choose the appropriate flow-control method for any specific traffic circle. That is, summarize the conditions under which each type of

5、traffic-control method should be used. When traffic lights are recommended, explain a method for determining how many seconds each light should remain green (which may vary according to the time of day and other factors). Illustrate how your model works with specific examples. 2009 MCM B: Energy and

6、 the Cell PhoneThis question involves the “energy” consequences of the cell phone revolution. Cell phone usage is mushrooming涌现, and many people are using cell phones and giving up their landline telephones. What is the consequence of this in terms of electricity use? Every cell phone comes with a b

7、attery and a recharger.充电器 Requirement 1 Consider the current US, a country of about 300 million people. Estimate from available data the number H of households, with m members each, that in the past were serviced by landlines. Now, suppose that all the landlines are replaced by cell phones; that is

8、, each of the m members of the household has a cell phone. Model the consequences of this change for electricity utilization使用 in the current US, both during the transition and during the steady state. The analysis should take into account the need for charging索价 the batteries of the cell phones, as

9、 well as the fact that cell phones do not last as long as landline phones (for example, the cell phones get lost and break). Requirement 2 Consider a second “Pseudo US”a country of about 300 million people with about the same economic status as the current US. However, this emerging新兴的 country has n

10、either landlines nor cell phones. What is the optimal最理想的 way of providing phone service to this country from an energy perspective观点? Of course, cell phones have many social consequences and uses that landline phones do not allow. A discussion of the broad明显的 and hidden隐藏的 consequences of having on

11、ly landlines, only cell phones, or a mixture of the two is welcomed. Requirement 3 Cell phones periodically定期的 need to be recharged.充电 However, many people always keep their recharger plugged 塞in. Additionally, many people charge their phones every night, whether they need to be recharged or not. Mo

12、del the energy costs of this wasteful practice for a Pseudo US based upon your answer to Requirement 2. Assume that the Pseudo US supplies electricity from oil. Interpret 解释your results in terms of 就而言barrels桶 of oil. Requirement 4 Estimates vary on the amount of energy that is used by various recha

13、rger types (TV, DVR, computer peripherals, and so forth) when left plugged in but not charging the device.充电器插着单没用来给手机充电 Use accurate data to model the energy wasted by the current US in terms of barrels of oil per day. Requirement 5 Now consider population and economic growth over the next 50 years

14、. How might a typical Pseudo US grow? For each 10 years for the next 50 years, predict the energy needs for providing phone service based upon your analysis in the first three requirements. Again, assume electricity is provided from oil. Interpret your predictions in term of barrels of oil. 2008 年美国

15、大学生数学建模竞赛 MCM、ICM 试题2008 MCM A: Take a BathConsider the effects on land from the melting of the north polar ice cap due to the predicted increase in global temperatures. Specifically, model the effects on the coast of Florida every ten years for the next 50 years due to the melting, with particular

16、attention given to large metropolitan areas. Propose appropriate responses to deal with this. A careful discussion of the data used is an important part of the answer. 2008 MCM B: Creating Sudoku数独 PuzzlesDevelop an algorithm演算法 to construct 构造Sudoku puzzles 困惑of varying difficulty. Develop metrics公

17、尺的 to define a difficulty level. The algorithm and metrics should be extensible 可扩充的to a varying number of difficulty levels. You should illustrate阐明 the algorithm with at least 4 difficulty levels. Your algorithm should guarantee a unique solution. Analyze the complexity复杂度 of your algorithm. Your

18、objective目标 should be to minimize the complexity of the algorithm and meet the above requirements. 2007 年美国大学生数学建模竞赛 MCM、ICM 试题2007 MCM A: Gerrymandering选区不公正的重新划分Gerrymandering The United States Constitution宪法 provides that the House of Representatives shall be composed of 由什么组成some number (current

19、ly 435) of individuals who are elected from each state in proportion比例 to the states population relative相应的，成比例的 to that of the country as a whole. While this provides a way of determining how many representatives 州议员each state will have, it says nothing about how the district行政区，辖区 represented by a

20、 particular 特定的representative shall be determined geographically. This oversight疏忽出错 has led to egregious 极坏的(at least some people think so, usually not the incumbent) district shapes 情况that look “unnatural” by some standards. 标准Hence the following question: Suppose you were given the opportunity to

21、 draw congressional美国国会的 districts for a state. How would you do so as a purely纯粹的 “baseline”基准线 exercise to create the “simplest” shapes for all the districts in a state? The rules include only that each district in the state must contain the same population. The definition of “simple” is up to you

22、; but you need to make a convincing 令人信服的argument论据 to voters选举人 in the state that your solution is fair公平的. As an application应用 of your method, draw 草拟制定geographically地理上的 simple congressional districts for the state of New York. 2007 MCM B: The Airplane Seating ProblemAirlines are free to seat pas

23、sengers waiting to board上飞机 an aircraft in any order whatsoever无论什么. It has become customary习惯上的 to seat passengers with special needs first, followed by first-class 头等的passengers (who sit at the front of the plane). Then coach and business-class passengers are seated by groups of rows, beginning wi

24、th the row at the back of the plane and proceeding行进 forward. Apart from consideration 考虑的事of the passengers wait time, from the airlines point of view, time is money, and boarding time is best minimized. The plane makes money for the airline only when it is in motion,运转 and long boarding times limi

25、t the number of trips航行 that a plane can make in a day. The development of larger planes, such as the Airbus A380 (800 passengers), accentuate更突出 the problem of minimizing boarding (and deboarding) time. Devise想出 and compare procedures步骤 for boarding and deboarding planes with varying numbers of pas

26、sengers: small (85210), midsize (210330), and large (450800). Prepare an executive 执行的summary, not to exceed超出 two single-spaced pages, in which you set out your conclusions to an audience of airline executives, gate agents,代理人 and flight crews. 工作人员An article appeared in the NY Times Nov 14, 2006 a

27、ddressing procedures currently当前 being followed and the importance to the airline of finding better solutions. The article can be seen at: 2006 年美国大学生数学建模竞赛 MCM、ICM 试题2006 MCM A: Positioning and Moving Sprinkler 洒水器Systems for Irrigation灌溉There are a wide variety of techniques available for irrigati

28、ng a field. The technologies range from advanced drip洒下 systems to periodic 周期性的flooding. One of the systems that is used on smaller ranches is the use of “hand move” irrigation systems. Lightweight 轻量的aluminum 铝pipes 输送管with sprinkler洒水器 heads are put in place across fields, and they are moved by h

29、and at periodic 周期性的intervals距离to insure that the whole field receives an adequate amount of water. This type of irrigation system is cheaper and easier to maintain 维修than other systems. It is also flexible, allowing for use on a wide variety of fields and crops. The disadvantage is that it requires

30、 a great deal of time and effort to move and set up the equipment at regular intervals. 距离Given that this type of irrigation system is to be used, how can it be configured 安装to minimize the amount of time required to irrigate a field that is 80 meters by 30 meters? For this task you are asked to fin

31、d an algorithm算法 to determine how to irrigate the rectangular矩形的 field that minimizes the amount of time required by a rancher农场工人 to maintain the irrigation system. One pipe输送管 set is used in the field. You should determine the number of sprinklers洒水器 and the spacing between sprinklers, and you sho

32、uld find a schedule计划表 to move the pipes, including where to move them. A pipe set consists of a number of pipes that can be connected together in a straight line. Each pipe has a 10 cm inner diameter直径 with rotating 转动spray喷雾器 nozzles 管嘴that have a 0.6 cm inner diameter. When put together the resulting pipe is 20 meters long. At the water source, the pressure is 420 Kilo- Pascals and has a flow rate of 150 liters per minute. No part of the field should receive more than 0.75 cm per hour of water, and each part of the field should receive at lea