美国数模竞赛题目历届.docx

1、美国数模竞赛题目历届2008 MCM ProblemsPROBLEM 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 parti

2、cular 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. PROBLEM B: Creating Sudoku PuzzlesDevelop an algorithm to construct Sudoku puzzles of varying difficulty. Develop metrics to d

3、efine a difficulty level. The algorithm and metrics should be extensible to a varying number of difficulty levels. You should illustrate the algorithm with at least 4 difficulty levels. Your algorithm should guarantee a unique solution. Analyze the complexity of your algorithm. Your objective should

4、 be to minimize the complexity of the algorithm and meet the above requirements. 2007 MCM ProblemsPROBLEM A: Gerrymandering The United States Constitution provides that the House of Representatives shall be composed of some number (currently 435) of individuals who are elected from each state in pro

5、portion to the states population relative to that of the country as a whole. While this provides a way of determining how many representatives each state will have, it says nothing about how the district represented by a particular representative shall be determined geographically. This oversight ha

6、s led to egregious (at least some people think so, usually not the incumbent) district shapes that look “unnatural” by some standards. Hence the following question: Suppose you were given the opportunity to draw congressional districts for a state. How would you do so as a purely “baseline” exercise

7、 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; but you need to make a convincing argument to voters in the state that your solution is fair. As an app

8、lication of your method, draw geographically simple congressional districts for the state of New York. PROBLEM B: The Airplane Seating ProblemAirlines are free to seat passengers waiting to board an aircraft in any order whatsoever. It has become customary to seat passengers with special needs first

9、, followed by first-class passengers (who sit at the front of the plane). Then coach and business-class passengers are seated by groups of rows, beginning with the row at the back of the plane and proceeding forward. Apart from consideration of the passengers wait time, from the airlines point of vi

10、ew, time is money, and boarding time is best minimized. The plane makes money for the airline only when it is in motion, and long boarding times limit the number of trips that a plane can make in a day. The development of larger planes, such as the Airbus A380 (800 passengers), accentuate the proble

11、m of minimizing boarding (and deboarding) time.Devise and compare procedures for boarding and deboarding planes with varying numbers of passengers: small (85210), midsize (210330), and large (450800).Prepare an executive summary, not to exceed two single-spaced pages, in which you set out your concl

12、usions to an audience of airline executives, gate agents, and flight crews.Note: The 2 page executive summary is to be included IN ADDITION to the reports required by the contest guidelines.An article appeared in the NY Times Nov 14, 2006 addressing procedures currently being followed and the import

13、ance to the airline of finding better solutions. The article can be seen at: 2006 MCM ProblemsPROBLEM A: Positioning and Moving Sprinkler Systems for IrrigationThere are a wide variety of techniques available for irrigating a field. The technologies range from advanced drip systems to periodic flood

14、ing. One of the systems that is used on smaller ranches is the use of hand move irrigation systems. Lightweight aluminum pipes with sprinkler heads are put in place across fields, and they are moved by hand at periodic intervals to insure that the whole field receives an adequate amount of water. Th

15、is type of irrigation system is cheaper and easier to maintain than other systems. It is also flexible, allowing for use on a wide variety of fields and crops. The disadvantage is that it requires a great deal of time and effort to move and set up the equipment at regular intervals.Given that this t

16、ype of irrigation system is to be used, how can it be configured 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 find an algorithm to determine how to irrigate the rectangular field that minimizes the amount of time required

17、by a rancher to maintain the irrigation system. One pipe set is used in the field. You should determine the number of sprinklers and the spacing between sprinklers, and you should find a schedule to move the pipes, including where to move them.A pipe set consists of a number of pipes that can be con

18、nected together in a straight line. Each pipe has a 10 cm inner diameter with rotating spray nozzles 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- Pascal抯 and has a flow rate of 150 liters per minute. No part

19、 of the field should receive more than 0.75 cm per hour of water, and each part of the field should receive at least 2 centimeters of water every 4 days. The total amount of water should be applied as uniformly as possible PROBLEM B: Wheel Chair Access at AirportsOne of the frustrations with air tra

20、vel is the need to fly through multiple airports, and each stop generally requires each traveler to change to a different airplane. This can be especially difficult for people who are not able to easily walk to a different flights waiting area. One of the ways that an airline can make the transition

21、 easier is to provide a wheel chair and an escort to those people who ask for help. It is generally known well in advance which passengers require help, but it is not uncommon to receive notice when a passenger first registers at the airport. In rare instances an airline may not receive notice from

22、a passenger until just prior to landing.Airlines are under constant pressure to keep their costs down. Wheel chairs wear out and are expensive and require maintenance. There is also a cost for making the escorts available.2005 MCM ProblemsPROBLEM A: Flood PlanningLake Murray in central South Carolin

23、a is formed by a large earthen dam, which was completed in 1930 for power production. Model the flooding downstream in the event there is a catastrophic earthquake that breaches the dam.Two particular questions:Rawls Creek is a year-round stream that flows into the Saluda River a short distance down

24、river from the dam. How much flooding will occur in Rawls Creek from a dam failure, and how far back will it extend?Could the flood be so massive downstream that water would reach up to the S.C. State Capitol Building, which is on a hill overlooking the Congaree River?PROBLEM B: Tollbooths Heavily-t

25、raveled toll roads such as the Garden State Parkway , Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas. Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disrupt

26、ion caused by the toll plazas. Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza. Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza, the flow of vehicles is re

27、quired to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza. Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza. When traffic is very heavy, congestion also builds at the entry to the toll plaza because o

28、f the time required for each vehicle to pay the toll. Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effe

29、ctive than the current practice? Note that the definition of optimal is up to you to determine.2005 ICM ProblemPROBLEM C: Nonrenewable ResourcesSelect a vital nonrenewable or exhaustible resource (water, mineral, energy, food, etc.) for which your team can find appropriate world-wide historic data o

30、n its endowment, discovery, annual consumption, and price.The modeling tasks are:1.Using the endowment, discoveries, and consumption data, model the depletion or degradation of the commodity over a long horizon using resource modeling principles. 2.Adjust the model to account for future economic, de

31、mographic, political and environmental factors. Be sure to reveal the details of your model, provide visualizations of the models output, and explain limitations of the model. 3.Create a fair, practical harvesting/management policy that may include economic incentives or disincentives, which sustain

32、 the usage over a long period of time while avoiding severe disruption of consumption, degradation or rapid exhaustion of the resource. 4.Develop a security policy that protects the resource against theft, misuse, disruption, and unnecessary degradation or destruction of the resource. Other issues that may need to be addressed are political and security management alternatives associated with these policies. 5.Develop policies to control any short- or long-term environmental effects of the harvesti