美国数学建模比赛历年试题.docx

1、美国数学建模比赛历年试题19852014年美国大学生数学建模竞赛题目集锦目录1985 MCM A: Animal Populations 31985 MCM B: Strategic Reserve Management 31986 MCM A: Hydrographic Data 41986 MCM B: Emergency-Facilities Location 41987 MCM A: The Salt Storage Problem 51987 MCM B: Parking Lot Design 51988 MCM A: The Drug Runner Problem 51988 MC

2、M B: Packing Railroad Flatcars 61989 MCM A: The Midge Classification Problem 61989 MCM B: Aircraft Queueing 61990 MCM A: The Brain-Drug Problem 61990 MCM B: Snowplow Routing 71991 MCM A: Water Tank Flow 81991 MCM B: The Steiner Tree Problem 81992 MCM A: Air-Traffic-Control Radar Power 81992 MCM B: E

3、mergency Power Restoration 91993 MCM A: Optimal Composting 101993 MCM B: Coal-Tipple Operations 111994 MCM A: Concrete Slab Floors 111994 MCM B: Network Design 121995 MCM A: Helix Construction 131995 MCM B: Faculty Compensation 131996 MCM A: Submarine Tracking 131996 MCM B: Paper Judging 131997 MCM

4、A: The Velociraptor Problem 141997 MCM B: Mix Well for Fruitful Discussions 151998 MCM A: MRI Scanners 161998 MCM B: Grade Inflation 171999 MCM A: Deep Impact 171999 MCM B: Unlawful Assembly 182000 MCM A: Air Traffic Control 182000 MCM B: Radio Channel Assignments 192001 MCM A: Choosing a Bicycle Wh

5、eel 202001 MCM B: Escaping a Hurricanes Wrath (An Ill Wind.) 212002 MCM A: Wind and Waterspray 232002 MCM B: Airline Overbooking 232003 MCM A: The Stunt Person 242003 MCM B: Gamma Knife Treatment Planning 242004 MCM A: Are Fingerprints Unique? 252004 MCM B: A Faster QuickPass System 252005 MCM A: Fl

6、ood Planning 262005 MCM B: Tollbooths 262006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation 272006 MCM B: Wheel Chair Access at Airports 272007 MCM A: Gerrymandering 292007 MCM B: The Airplane Seating Problem 292008 MCM A: Take a Bath 302008 MCM B: Creating Sudoku Puzzles 302009 MCM

7、A: Designing a Traffic Circle 302009 MCM B: Energy and the Cell Phone 302010 MCM A: The Sweet Spot 322010 MCM B: Criminology 322011 MCM A: Snowboard Course 332011 MCM B: Repeater Coordination 332012 MCM A: The Leaves of a Tree 332012 MCM B: Camping along the Big Long River 342013 MCM A: The Ultimate

8、 Brownie Pan 342013 MCM B: Water, Water, Everywhere 352014 MCM A:The Keep-Right-Except-To-Pass Rule 352014 MCM B:College Coaching Legends 361985 MCM A: Animal PopulationsChoose a fish or mammal for which appropriate data are available to model it accurately. Model the animals natural interactions wi

9、th its environment by expressing population levels of different groups in terms of the significant parameters of the environment. Then adjust the model to account for harvesting in a form consistent with the actual method by which the animal is harvested. Include any outside constraints imposed by f

10、ood or space limitations that are supported by the data.Consider the value of the various quantities involved, the number harvested, and the population size itself, in order to devise a numerical quantity that represents the overall value of the harvest. Find a harvesting policy in terms of populati

11、on size and time that optimizes the value of the harvest over a long period of time. Check that the policy optimizes that value over a realistic range of environmental conditions.1985 MCM B: Strategic Reserve ManagementCobalt, which is not produced in the US, is essential to a number of industries.

12、(Defense accounted for 17% of the cobalt production in 1979.) Most cobalt comes from central Africa, a politically unstable region. The Strategic and Critical Materials Stockpiling Act of 1946 requires a cobalt reserve that will carry the US through a three-year war. The government built up a stockp

13、ile in the 1950s, sold most of it off in the early 1970s, and then decided to build it up again in the late 1970s, with a stockpile goal of 85.4 million pounds. About half of this stockpile had been acquired by 1982.Build a mathematical model for managing a stockpile of the strategic metal cobalt. Y

14、ou will need to consider such questions as: How big should the stockpile be? At what rate should it be acquired? What is a reasonable price to pay for the metal?You will also want to consider such questions as: At what point should the stockpile be drawn down? At what rate should it be drawn down? A

15、t what price is it reasonable to sell the metal? How should it be allocated?Useful Information on CobaltThe government has projected a need ot 25 million pounds of cobalt in 1985.The U.S. has about 100 million pounds of proven cobalt deposits. Production becomes economically feasible when the price

16、reaches $22/lb (as occurred in 1981). It takes four years to get operations rolling, and thsn six million pounds per year can be produced.In 1980, 1.2 million pounds of cobalt were recycled, 7% of total consumption.1986 MCM A: Hydrographic DataThe table below gives the depth Z of water in feet for s

17、urface points with rectangular coordinates X, Y in yards table of 14 data points omitted. The depth measurements were taken at low tide. Your ship has a draft of five feet. What region should you avoid within the rectangle (75,200) x (-50, 150)?XYZ129.07.54140.0141.58108.528.0688.0147.08185.522.5619

18、5.0137.58105.585.58157.5-6.59107.5-81.0977.03.08162.0-66.59162.084.04117.5-35.591986 MCM B: Emergency-Facilities LocationThe township of Rio Rancho has hitherto not had its own emergency facilities. It has secured funds to erect two emergency facilities in 1986, each of which will combine ambulance,

19、 fire, and police services. Figure 1 indicates the demand figure omitted, or number of emergencies per square block, for 1985. The “L” region in the north is an obstacle, while the rectangle in the south is a part with shallow pond. It takes an emergency vehicle an average of 15 seconds to go one bl

20、ock in the N-S direction and 20 seconds in the E-W direction. Your task is to locate the two facilities so as to minimize the total response time. Assume that the demand is concentrated at the center of the block and that the facilities will be located on corners. Assume that the demand is uniformly

21、 distributed on the streets bordering each block and that the facilities may be located anywhere on the streets.1987 MCM A: The Salt Storage ProblemFor approximately 15 years, a Midwestern state has stored salt used on roads in the winter in circular domes. Figure 1 shows how salt has been stored in

22、 the past. The salt is brought into and removed from the domes by driving front-end loaders up ramps of salt leading into the domes. The salt is piled 25 to 30 ft high, using the buckets on the front-end loaders.Recently, a panel determined that this practice is unsafe. If the front-end loader gets

23、too close to the edge of the salt pile, the salt might shift, and the loader could be thrown against the retaining walls that reinforce the dome. The panel recommended that if the salt is to be piled with the use of the loaders, then the piles should be restricted to a matimum height of 15 ft.Constr

24、uct a mathematical model for this situation and find a recommended maximum height for salt in the domes.1987 MCM B: Parking Lot DesignThe owner of a paved, 100 by 200 , corner parking lot in a New England town hires you to design the layout, that is, to design how the lines are to be painted.You rea

25、lize that squeezing as many cars into the lot as possible leads to right-angle parking with the cars aligned side by side. However, inexperienced drivers have difficulty parking their cars this way, which can give rise to expensive insurance claims. To reduce the likelihood of damage to parked vehic

26、les, the owner might then have to hire expert drivers for valet parking.On the other hand, most drivers seem to have little difficulty in parking in one attempt if there is a large enough turning radius from the access lane. Of course, the wider the access lane, the fewer cars can be accommodated in

27、 the lot, leading to less revenue for the parking lot owner.1988 MCM A: The Drug Runner ProblemTwo listening posts 5.43 miles apart pick up a brief radio signal. The sensing devices were oriented at 110 degrees and 119 degrees, respectively, when the signal was detected; and they are accurate to wit

28、hin 2 degrees. The signal came from a region of active drug exchange, and it is inferred that there is a powerboat waiting for someone to pick up drugs. it is dusk, the weather is calm, and there are no currents. A small helicopter leaves from Post 1 and is able to fly accurately along the 110 degre

29、e angle direction. The helicopters speed is three times the speed of the boat. The helicopter will be heard when it gets within 500 ft of the boat. This helicopter has only one detection device, a searchlight. At 200 ft, it can just illuminate a circular region with a radius of 25 ft. Develop an opt

30、imal search method for the helicopter. Use a 95% confidence level in your calculations.1988 MCM B: Packing Railroad FlatcarsTwo railroad flatcars are to be loaded with seven types of packing crates. The crates have the same width and height but varying thickness (t, in cm) and weight (w, in kg). Tab

31、le 1 gives, for each crate, the thickness, weight, and number available table omitted. Each car has 10.2 meters of length available for packing the crates (like slices of toast) and can carry up to 40 metric tons. There is a special constraint on the total number of C_5, C_6, and C_7 crates because

32、of a subsequent local trucking restriction: The total space (thickness) occupied by these crates must not exceed 302.7 cm. Load the two flatcars (see Figure 1) so as to minimize the wasted floor space figure omitted.1989 MCM A: The Midge Classification ProblemTwo species of midges, Af and Apf, have been identified by biologists Grogan and Wirth on the basis of antenna