美赛历年题目.docx

1、美赛历年题目具体可以登录官网看2014 MCM ProblemsPROBLEM A:The Keep-Right-Except-To-Pass RuleIn countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employa rule tha

2、t requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane.Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wi

3、sh to examine tradeoffs between traffic flow and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this rule effective in promoting better traffic flow?

4、 If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.In countries where driving automobiles on the left is the norm, argue whether or not your solution can be carried

5、over with a simple change of orientation, or would additional requirements be needed.Lastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system either part of the road network or imbed

6、ded in the design of all vehicles using the roadway to what extent would this change the results of your earlier analysis?PROBLEM B:College Coaching LegendsSports Illustrated, a magazine for sports enthusiasts, is looking for the “best all time college coach” male or female for the previous century.

7、 Build a mathematical model to choose thebestcollege coach or coaches (past or present) from among either male or female coaches in such sports as college hockey or field hockey, football, baseball or softball, basketball, or soccer. Does it make a difference which time line horizon that you use in

8、your analysis, i.e., does coaching in 1913 differ from coaching in 2013? Clearly articulate your metrics for assessment. Discuss how your model can be applied in general across both genders and all possible sports. Present your models top 5 coaches in each of 3 different sports.In addition to the MC

9、M format and requirements, prepare a 1-2 page article forSports Illustratedthat explains your results and includes a non-technical explanation of your mathematical model thatsports fanswill understand.Problem AProblem: Unloading Commuter TrainsTrains arrive often at a central Station, the nexus for

10、many commuter trains from suburbs of larger cities on a “commuter” line. Most trains are long (perhaps 10 or more cars long). The distance a passenger has to walk to exit the train area is quite long. Each train car has only two exits, one near each end so that the cars can carry as many people as p

11、ossible. Each train car has a center aisle and there are two seats on one side and three seats on the other for each row of seats.To exit a typical station of interest, passengers must exit the car, and then make their way to a stairway to get to the next level to exit the station. Usually these tra

12、ins are crowded so there is a “fan” of passengers from the train trying to get up the stairway. The stairway could accommodate two columns of people exiting to the top of the stairs.Most commuter train platforms have two tracks adjacent to the platform. In the worst case, if two fully occupied train

13、s arrived at the same time, it might take a long time for all the passengers to get up to the main level of the station.Build a mathematical model to estimate the amount of time for a passenger to reach the street level of the station to exit the complex. Assume there arencars to a train, each car h

14、as lengthd. The length of the platform isp, and the number of stairs in each staircase isq.Use your model to specifically optimize (minimize) the time traveled to reach street level to exit a station for the following:Requirement 1.One fully occupied trains passengers to exit the train, and ascend t

15、he stairs to reach the street access level of the stationRequirement 2.Two fully occupied trains passengers (all passengers exit onto a common platform) to exit the trains, and ascend the stairs to reach the street access level of the station.Requirement 3.If you could redesign the location of the s

16、tairways along the platform, where should these stairways be placed to minimize the time for one or two trains passengers to exit the station?Requirement 4.How does the time to street level vary with the number s of stairways that one builds?Requirement 5.How does the time vary if the stairways can

17、accommodatekpeople,kan integer greater than one?In addition to the HiMCM format, prepare a short non-technical article to the director of transportation explaining why they should adopt your model to improve exiting a station.Problem BProblem: The Next Plague?In 2014, the world saw the infectious Eb

18、ola virus spreading in western Africa. Throughout human history, epidemics have come and gone with some infecting and/or killing thousands and lasting for years and others taking less of a human toll. Some believe these events are just natures way of controlling the growth of a species while others

19、think they could be a conspiracy or deliberate act to cause harm. This problem will most likely come down to how to expend (or not expend) scarce resources (doctors, containment facilities, money, research, serums, etc) to deal with a crisis.Situation: A routine humanitarian mission on an island in

20、Indonesia reported a small village where almost half of its 300 inhabitants are showing similar symptoms. In the past week, 15 of the “infected” have died. This village is known to trade with nearby villages and other islands. Your modeling team works for a major center of disease control in the cap

21、ital ofyourcountry (or if you prefer, for the International World Health Organization).Requirement 1:Develop a mathematical model(s) that performs the following functions as well as how/when to best allocate these scarce resources and Determines and classifies the type and severity of the spread of

22、the disease Determines if an epidemic is contained or not Triggers appropriate measures (when to treat, when to transport victims, when to restrict movement, when to let a disease run its course, etc) to contain a disease Note: While you may want to start with the well-known “SIR” family of models f

23、or parts of this problem, consider others, modifications to the SIR, multiple models, or creating your own.Requirement 2:Based on the information given, your model, and the assumptions your team has made, what initial recommendations does your team have for your countrys center for disease control?

24、(Give 3-5 recommendations with justifications)Additional Situational Information: A multi-national research team just returned to your countrys capital after spending 7 days gathering information in the infected village.Requirement 3:You can ask them up to 3 questions to improve your model. What wou

25、ld you ask and why?Additional Situational Information: The multi-national research team concluded that the disease: Appears to spread through contact with bodily fluids of an infected person The elderly and children are more likely to die if infected A nearby island is starting to show similar signs

26、 of infection One of the researchers that returned to your capital appears infectedRequirement 4:How does the additional information above change/modify your model?Requirement 5:Write a one-page synopsis of your findings for your local non-technical news outlet.2013 MCM ProblemsPROBLEM A:The Ultimat

27、e Brownie PanWhen baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges). In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. However,

28、since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven. Develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between.Assume1. A width

29、to length ratio ofW/L for the oven which is rectangular in shape.2. Each pan must have an area ofA.3. Initially two racks in the oven, evenly spaced.Develop a model that can be used to select the best type of pan (shape) under the following conditions:1. Maximize number of pans that can fit in the o

30、ven (N)2. Maximize even distribution of heat (H) for the pan3. Optimize a combination of conditions (1) and (2) where weights p and (1-p) are assigned to illustrate how the results vary with different values ofW/Landp.In addition to your MCM formatted solution, prepare a one to two page advertising

31、sheet for the new Brownie Gourmet Magazine highlighting your design and results.PROBLEM B:Water, Water, EverywhereFresh water is the limiting constraint for development in much of the world. Build a mathematical model for determining an effective, feasible, and cost-efficient water strategy for 2013

32、 to meet the projected water needs of pick one country from the list below in 2025, and identify the best water strategy. In particular, your mathematical model must address storage and movement; de-salinization; and conservation. If possible, use your model to discuss the economic, physical, and environmental implications of your strategy. Provide a non-technical position paper to g