1987B 数学建模MCM题目

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1987年美国大学生数学建模竞赛试题

MCM1987A 盐的储存问题
大约15年以来，美国中西部的一个州一直把用于冬天洒在马路上的盐储存在球形屋顶的仓库里，图A-5表示了过去盐是怎样储存的，在用盐铺成的坡道上通过驾驶平头铲车把盐运进、运出仓库，用平头铲车上的铲斗把盐堆成25~30英尺高。

最近一个小组认为这种做法是不安全的，如果铲车太靠近盐堆的顶端，盐就要滑动，铲车就会翻到为加固仓库而筑的拥壁上去，小组建议，如果盐堆是用铲车堆起来的，那么盐堆最高不要超过5英尺。

对这种情况建立一个数学模型，并求出仓库内盐堆的最大高度。

MCM1987B 停车场问题
在New England(新英格兰，美国东北部一地区)一个镇上位于街角处，有一个100英尺×200英尺的停车场，场主雇你来设计这个停车场，也就是如何在停车场的地上画线。

你可能认为要把尽可能多的车驶进停车场，一定应该一辆挨一辆地直角停放，但是缺乏经验的司机感到这样停放有困难，会引起昂贵的保险费要求，为了减少停放车辆时可能造成的损坏，场主就要雇用一些专门停放汽车的有经验的司机。

另一方面，如果汽车从通道进来有一个足够大的转弯半径，那么大多数司机都能轻而易举地一次停放成功。

当然，通道越宽，能够容纳的车辆越少，这会导致停车场场主收入的减少。

美国数学建模竞赛1985-2013试题

1985~2013年美国大学生数学建模竞赛题目集锦目录1985 MCM A: Animal Populations (3)1985 MCM B: Strategic Reserve Management (3)1986 MCM A: Hydrographic Data (4)1986 MCM B: Emergency-Facilities Location (4)1987 MCM A: The Salt Storage Problem (5)1987 MCM B: Parking Lot Design (5)1988 MCM A: The Drug Runner Problem (5)1988 MCM B: Packing Railroad Flatcars (6)1989 MCM A: The Midge Classification Problem (6)1989 MCM B: Aircraft Queueing (6)1990 MCM A: The Brain-Drug Problem (6)1990 MCM B: Snowplow Routing (7)1991 MCM A: Water Tank Flow (8)1991 MCM B: The Steiner Tree Problem (8)1992 MCM A: Air-Traffic-Control Radar Power (8)1992 MCM B: Emergency Power Restoration (9)1993 MCM A: Optimal Composting (10)1993 MCM B: Coal-Tipple Operations (11)1994 MCM A: Concrete Slab Floors (11)1994 MCM B: Network Design (12)1995 MCM A: Helix Construction (13)1995 MCM B: Faculty Compensation (13)1996 MCM A: Submarine Tracking (13)1996 MCM B: Paper Judging (13)1997 MCM A: The Velociraptor Problem (14)1997 MCM B: Mix Well for Fruitful Discussions (15)1998 MCM A: MRI Scanners (16)1998 MCM B: Grade Inflation (17)1999 MCM A: Deep Impact (17)1999 MCM B: Unlawful Assembly (18)2000 MCM A: Air Traffic Control (18)2000 MCM B: Radio Channel Assignments (19)2001 MCM A: Choosing a Bicycle Wheel (20)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...). (21)2002 MCM A: Wind and Waterspray (23)2002 MCM B: Airline Overbooking (23)2003 MCM A: The Stunt Person (24)2003 MCM B: Gamma Knife Treatment Planning (24)2004 MCM A: Are Fingerprints Unique? (25)2004 MCM B: A Faster QuickPass System (25)2005 MCM A: Flood Planning (26)2005 MCM B: Tollbooths (26)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation (27)2006 MCM B: Wheel Chair Access at Airports (28)2007 MCM A: Gerrymandering (29)2007 MCM B: The Airplane Seating Problem (29)2008 MCM A: Take a Bath (30)2008 MCM B: Creating Sudoku Puzzles (30)2009 MCM A: Designing a Traffic Circle (30)2009 MCM B: Energy and the Cell Phone (30)2010 MCM A: The Sweet Spot (32)2010 MCM B: Criminology (32)2011 MCM A: Snowboard Course (33)2011 MCM B: Repeater Coordination (33)2012 MCM A: The Leaves of a Tree (33)2012 MCM B: Camping along the Big Long River (34)2013 MCM A: The Ultimate Brownie Pan (34)2013 MCM B: Water, Water, Everywhere (35)1985 MCM A: Animal PopulationsChoose a fish or mammal for which appropriate data are available to model it accurately. Model the animal's natural interactions with 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 food 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 population 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. (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 stockpile 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. You 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?▪At 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 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 surface 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)?1986 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, 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 block 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 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 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 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.Construct 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 realize 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 vehicles, 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 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 within 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 degree angle direction. The helicopter's 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 optimal 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). Table 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 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 and wing length (see Figure 1). It is important to be able to classify a specimen as Af of Apf, given the antenna and wing length.1. Given a midge that you know is species Af or Apf, how would you go about classifying it?2. Apply your method to three specimens with (antenna, wing) lengths(1.24,1.80),(1.28,1.84),(1.40,2.04).3. Assume that the species is a valuable pollinator and species Apf is a carrier of adebilitating disease. Would you modify your classification scheme and if so, how?1989 MCM B: Aircraft QueueingA common procedure at airports is to assign aircraft (A/C) to runways on a first-come-first-served basis. That is, as soon as an A/C is ready to leave the gate (―push-back‖), the pilot calls ground control and is added to the queue. Suppose that a control tower has access to a fast online database with the following information for each A/C:▪the time it is scheduled for pushback;▪the time it actually pushes back; the number of passengers who are scheduled to make a connection at the next stop, as well as the time to make that connection; and▪the schedule time of arrival at its next stop Assume that there are seven types of A/C with passenger capacities varying from 100 to 400 in steps of 50. Develop and analyze amathematical model that takes into account both the travelers' and airlines' satisfaction.1990 MCM A: The Brain-Drug ProblemResearches on brain disorders test the effects of the new medical drugs – for example, dopamine against Parkinson's disease – with intracerebral injections. To this end, they must estimate the size and the sape of the spatial distribution of the drug after the injection, in order to estimate accurately the region of the brain that the drug has affected.The research data consist of the measurements of the amounts of drug in each of 50 cylindrical tissue samples (see Figure 1 and Table 1). Each cylinder has length 0.76 mm and diameter 0.66 mm. The centers of the parallel cylinders lie on a grid with mesh 1mm X 0.76mm X 1mm, so that the sylinders touch one another on their circular bases but not along their sides, as shown in the accompanying figure. The injection was made near the center of the cylinder with the highest scintillation count. Naturally, one expects that there is a drug also between the cylinders and outside the region covered by the samples.Estimate the distribution in the region affected by the drug.One unit represents a scintillation count, or 4.753e-13 mole of dopamine. For example, the table shows that the middle rear sylinder contails 28353 units.Table 1. Amounts of drug in each of 50 cylindrical tissue samples.Rear vertical sectionFront vertical section1990 MCM B: Snowplow RoutingThe solid lines of the map (see Figure 1) represent paved two-lane county roads in a snow removal district in Wicomico County, Maryland [figure omitted]. The broken lines are state highways. After a snowfall, two plow-trucks are dispatched from a garage that is about 4 miles west of each of the two points (*) marked on the map. Find an efficient way to use the two trucks to sweep snow from the county roads. The trucks may use the state highways to access the county roads. Assume that the trucks neither break down nor get stuck and that the road intersections require no special plowing techniques.1991 MCM A: Water Tank FlowSome state water-right agencies require from communities data on the rate of water use, in gallons per hour, and the total amount of water used each day. Many communities do not have equipment to measure the flow of water in or out of the municipal tank. Instead, they can measure only the level of water in the tank, within 0.5% accuracy, every hour. More importantly, whenever the level in the tank drops below some minimum level L, a pump fills the tank up to the maximum level, H; however, there is no measurement of the pump flow either. Thus, one cannot readily relate the level in the tank to the amount of water used while the pump is working, which occurs once or twice per day, for a couple of hours each time. Estimate the flow out of the tank f(t) at all times, even when the pump is working, and estimate the total amount of water used during the day. Table 1 gives real data, from an actual small town, for one day[ table omitted]. The table gives the time, in, since the first measurement, and the level of water in the tank, in hundredths of a foot. For example, after 3316 seconds, the depth of water in the tank reached 31.10 feet. The tank is a vertical circular cylinder, with a height of 40 feet and a diameter of 57 feet. Usually, the pump starts filling the tank when the level drops to about 27.00 feet, and the pump stops when the level rises back to about 35.50 feet.1991 MCM B: The Steiner Tree ProblemThe cost for a communication line between two stations is proportional to the length of the line. The cost for conventional minimal spanning trees of a set of stations can often be cut by introducing―phantom‖ stations and then constructing a new Steiner tree. This device allows costs to be cut by up to 13.4% (= 1- sqrt(3/4)). Moreover, a network with n stations never requires more than n-2 points to construct the cheapest Steiner tree. Two simple cases are shown in Figure 1.For local networks, it often is necessary to use rectilinear or ―checker-board‖ distances, instead of straight Euclidean lines. Distances in this metric are computed as shown in Figure 2.Suppose you wish to design a minimum costs spanning tree for a local network with 9 stations. Their rectangular coordinates are: a(0,15), b(5,20), c(16,24), d(20,20), e(33,25), f(23,11), g(35,7), h(25,0) i(10,3). You are restricted to using rectilinear lines. Moreover, all ―phantom‖ stations must be located at lattice points (i.e., the coordinates must be integers). The cost for each line is its length.1. Find a minimal cost tree for the network.2. Suppose each stations has a cost w*d^(3/2), where d=degree of the station. If w=1.2, find aminimal cost tree.3. Try to generalize this problem1992 MCM A: Air-Traffic-Control Radar PowerYou are to determine the power to be radiated by an air-traffic-control radar at a major metropolitan airport. The airport authority wants to minimize the power of the radar consistent with safety andcost. The authority is constrained to operate with its existing antennae and receiver circuitry. The only option that they are considering is upgrading the transmitter circuits to make the radar more powerful. The question that you are to answer is what power (in watts) must be released by the radar to ensure detection of standard passenger aircraft at a distance of 100 kilometers.1992 MCM B: Emergency Power RestorationPower companies serving coastal regions must have emergency response systems for power outages due to storms. Such systems require the input of data that allow the time and cost required for restoration to be estimated and the ―value‖ of the outage judged by objective criteria. In the past, Hypothetical Electric Company (HECO) has been criticized in the media for its lack of a prioritization scheme.You are a consultant to HECO power company. HECO possesses a computerized database with real time access to service calls that currently require the following information:▪time of report,▪type of requestor,▪estimated number of people affected, and▪location (x,y).Cre sites are located at coordinates (0,0) and (40,40), where x and y are in miles. The region serviced by HECO is within -65 < x < 60 and -50 < y < 50. The region is largely metropolitan with an excellent road network. Crews must return to their dispatch site only at the beginning and end of shift. Company policy requires that no work be initiated until the storm leaves the area, unless the facility is a commuter railroad or hospital, which may be processed immediately if crews are available.HECO has hired you to develop the objective criteria and schedule the work for the storm restoration requirements listed in Table 1 using their work force described in Table 2. Note that the first call was received at 4:20 A.M. and that the storm left the area at 6:00 A.M. Also note that many outages were not reported until much later in the day.HECO has asked for a technical report for their purposes and an ―executive summary‖ i n laymen's terms that can be presented to the media. Further, they would like recommendations for the future. To determine your prioritized scheduling system, you will have to make additional assumptions. Detail those assumptions. In the future, you may desire additional data. If so, detail the information desired.Table 1. Storm restoration requirements. (table incomplete)Table 2. Crew descriptions.1993 MCM A: Optimal CompostingAn environmentally conscious institutional cafeteria is recycling customers' uneaten food into compost by means of microorganisms. Each day, the cafeteria blends the leftover food into a slurry, mixes the slurry with crisp salad wastes from the kitchen and a small amount of shredded newspaper, and feeds the resulting mixture to a culture of fungi and soil bacteria, which digest slurry, greens, and papers into usable compost. The crisp green provide pockets of oxygen for the fungi culture, and the paper absorbs excess humidity. At times, however, the fungi culture is unable or unwilling to digest as much of the leftovers as customers leave; the cafeteria does not blame the chef for the fungi culture's lack of appetite. Also, the cafeteria has received offers for the purchase of large quantities of it compost. Therefore, the cafeteria is investigating ways to increase its production of compost. Since it cannot yet afford to build a new composting facility, the cafeteria seeks methods to accelerate the fungi culture's activity, for instance, by optimizing the fungiculture's environment (currently held at about 120 F and 100% humidity), or by optimizing the composition of the moisture fed to the fungi culture, or both.Determine whether any relation exists between the proportions of slurry, greens, and paper in the mixture fed to the fungi culture, and the rate at which the fungi culture composts the mixture. if no relation exists, state so. otherwise, determine what proportions would accelerate the fungi culture's activity. In addition to the technical report following the format prescribed in the contest instructions, provide a one-page nontechnical recommendation for implementation for the cafeteria manager. Table 1 shows the composition of various mixtures in pounds of each ingredient kept in separate bins, and the time that it took the fungi to culture to compost the mixtures, from the date fed to the date completely composted [table omitted].1993 MCM B: Coal-Tipple OperationsThe Aspen-Boulder Coal Company runs a loading facility consisting of a large coal tipple. When the coal trains arrive, they are loaded from the tipple. The standard coal train takes 3 hours to load, and the tipple's capacity is 1.5 standard trainloads of coal. Each day, the railroad sends three standard trains to the loading facility, and they arrive at any time between 5 A.M. and 8 P.M. local time. Each of the trains has three engines. If a train arrives and sits idle while waiting to be loaded, the railroad charges a special fee, called a demurrage. The fee is $5,000 per engine per hour. In addition, a high-capacity train arrives once a week every Thursday between 11 A.M. and 1 P.M. This special train has five engines and holds twice as much coal as a standard train. An empty tipple can be loaded directly from the mine to its capacity in six hours by a single loading crew. This crew (and its associated equipment) cost $9,000 per hour. A second crew can be called out to increase the loading rate by conducting an additional tipple-loading operation at the cost of $12,000 per hour. Because of safety requirements, during tipple loading no trains can be loaded. Whenever train loading is interrupted to load the tipple, demurrage charges are in effect.The management of the Coal Company has asked you to determine the expected annual costs of this tipple's loading operations. Your analysis should include the following considerations:▪How often should the second crew be called out?▪What are the expected monthly demurrage costs?▪If the standard trains could be scheduled to arrive at precise times, what daily schedule would minimize loading costs? Would a third tipple-loading crew at $12,000 per hour reduce annual operations costs?▪Can this tipple support a fourth standard train every day?1994 MCM A: Concrete Slab FloorsThe U.S. Dept. of Housing and Urban Development (HUD) is considering constructing dwellings of various sizes, ranging from individual houses to large apartment complexes. A principal concern is to minimize recurring costs to occupants, especially the costs of heating and cooling. The region inwhich the construction is to take place is temperate, with a moderate variation in temperature throughout the year.Through special construction techniques, HUD engineers can build dwellings that do not need to rely on convection- that is, there is no need to rely on opening doors or windows to assist in temperature variation. The dwellings will be single-story, with concrete slab floors as the only foundation. You have been hired as a consultant to analyze the temperature variation in the concrete slab floor to determine if the temperature averaged over the floor surface can be maintained within a prescribed comfort zone throughout the year. If so, what size/shape of slabs will permit this?Part 1, Floor Temperature: Consider the temperature variation in a concrete slab given that the ambient temperature varies daily within the ranges given Table 1. Assume that the high occurs at noon and the low at midnight. Determine if slabs can be designed to maintain a temperature averaged over the floor surface within the prescribed comfort zone considering radiation only. Initially, assume that the heat transfer into the dwelling is through the exposed perimeter of the slab and that the top and bottom of the slabs are insulated. Comment on the appropriateness and sensitivity of these assumptions. If you cannot find a solution that satisfies Table 1, can you find designs that satisfy a Table 1 that you propose?Part 2, Building Temperature: Analyze the practicality of the initial assumptions and extend the analysis to temperature variation within the single-story dwelling. Can the house be kept within the comfort zone?Part 3, Cost of Construction: Suggest a design that considers HUD's objective of reducing or eliminating heating and cooling costs, considering construction restrictions and costs.1994 MCM B: Network DesignIn your company, information is shared among departments on a daily basis. This information includes the previous day's sales statistics and current production guidance. It is important to get this information out as quickly as possible. [Network diagram (with 5 nodes and 7 capacitated edges) omitted.]We are interested in scheduling transfers in an optimal way to minimize the total time it takes to complete them all. This minimum total time is called the makespan. Consider the three following situations for your company: [Three more network diagrams (on roughly 20 nodes each) omitted.]1995 MCM A: Helix ConstructionA small biotechnological company must design, prove, program and test a mathematical algorithm to locate ―in real time‖ all the intersections of a helix and a plane in general positions in space. Design, justify, program and test a method to compute all the intersections of a plane and a helix, both in general positions (at any locations and with any orientations) in space. A segment of the helix may represent, for example, a helicoidal suspension spring or a piece of tubing in a chemical or medical apparatus. Theoretical justification of the proposed algorithm is necessary to verify the solution from several points of view, for instance, through mathematical proofs of parts of the algorithm, and through tests of the final program with known examples. Such documentation and tests will be required by government agencies for medical use.1995 MCM B: Faculty CompensationAluacha Balaclava College, and undergraduate facility, has just hired a new Provost whose first priority is the institution of a fair and reasonable faculty-compensation plan. She has hired your consulting team to design a compensation system that reflects the following circumstances and principles: [Three paragraphs of details omitted] Design a new pay system, first withoutcost-of-living increases. Incorporate cost-of-living increases, and then finally, design a transition process for current faculty that will move all salaries towards your system without reducing anyone's salary. The Provost requires a detailed compensation system plan for implementation, as well as a brief, clear, executive summary outlining the model, its assumptions, strengths, weaknesses and expected results, which she can present to the Board and faculty. [A detailed table of current salaries is omitted.]1996 MCM A: Submarine TrackingThe world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large maving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, develop a method for detecting the presence of a moving submarine, its speed, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed frequency and amplitude.1996 MCM B: Paper JudgingWhen determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Let's say there are P=100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest contrains both the number of judges that can be obtained and the amount of time they can judge. For example if P=100, then J=8 is typical.。

PCMM竞赛试题

PCMM竞赛试题一、选择题（每题2分，共10分）1. 某工厂需要生产一批零件，每生产一个零件的成本是C元，如果生产N个零件，总成本是多少？A. N*CB. C/NC. N/CD. C+N2. 如果一个圆的半径是r，那么它的面积A是多少？A. πrB. πr²C. r²D. πr³3. 一个函数f(x)=2x+3，当x=1时，f(x)的值是多少？A. 5B. 6C. 7D. 84. 以下哪个是线性方程的标准形式？A. y = 3x + 2B. y = 3x² + 2C. x + y = 5D. x² + y = 55. 一个数列1, 3, 5, 7, ...，这个数列的第10项是多少？A. 17B. 19C. 21D. 23二、简答题（每题5分，共20分）1. 解释什么是二进制数，并给出十进制数25转换为二进制数的过程。

2. 描述如何使用勾股定理来解决一个直角三角形的问题。

3. 给出一个例子，说明如何使用概率论来预测一个事件的发生。

4. 解释什么是线性回归，并简述其在数据分析中的应用。

三、应用题（每题15分，共30分）1. 一个农场主想要知道他的农场面积。

他测量了农场的一边长为100米，另一边长为80米。

如果他想估算农场的面积，他应该如何做？请给出计算过程和结果。

2. 一家公司想要预测下个季度的销售额。

他们有过去四个季度的销售额数据：第一季度为100万美元，第二季度为120万美元，第三季度为130万美元，第四季度为140万美元。

请使用这些数据来预测下个季度的销售额，并解释你的预测方法。

四、论文题（每题40分，共40分）1. 选择一个你感兴趣的数学问题或数学概念，写一篇不少于500字的论文。

论文应包括问题的背景、问题的陈述、解决方案的分析以及结论。

注意：请在规定时间内完成所有题目，并确保你的答案清晰、准确。

祝你在PCMM竞赛中取得优异成绩！。

数学建模cumcm历届竞赛赛题基本解法

数学建模（cumcm）历届竞赛赛题基本解法来源：蒋冰的日志赛题解法一些不必须用到的算法92A施肥效果分析回归分析，因子分析，相关分析，参数估计92B蛋白质氨基酸的组合问题线性不定方程式,离散最优化93A非线性交调的频率设计拟合、规划93B足球队排名图论、层次分析、整数规划94A逢山开路图论、插值、动态规划线路设计，局部最优化，层次分析法94B锁具装箱问题图论、组合数学95A飞行管理问题非线性规划、线性规划能量梯度算法，线性规划，非线性规划，逐步逼近搜索，95B天车与冶炼炉的作业调度动态规划、排队论、图论petri网，随机性分析96A最优捕鱼策略微分方程、优化96B节水洗衣机非线性规划Gordon-Schaefer模型，97A零件的参数设计非线性规划敏感度分析、敏感度分析，统计检验，因素交替法，一维搜索，穷举法，随机模拟（MonterCarol)，模拟退火，最优速降法，97B截断切割的最优排列随机模拟、图论分支限界法，贪婪算法，最短路径（Dijkstra),启发式搜索（A*算法）98A一类投资组合问题多目标优化、非线性规划投资组合模型，灵敏度分析，多目标决策模型，偏好系数加权法，模糊线性规划法，多目标优化问题，随机投点法，98B灾情巡视的最佳路线图论、组合优化最小hamilton回路，最优旅行商路线99A自动化车床管理随机优化、计算机模拟D检验方法，99B钻井布局0-1规划、图论强局算法，0-1规划，全局搜索算法00ADNA序列分类模式识别、Fisher判别、人工神经网络广度优先，最小二乘法，欧氏距离vs 马氏距离，fisher分类法，人工神经网络（感知机模型，多层感知机，LVQ 矢量量化），隐马尔科夫模型，同源比较算法，傅立叶分析，动态规划，00B钢管订购和运输组合优化、运输问题线性规划，模拟退火，伏格尔法01A血管三维重建曲线拟合、曲面重建快速傅立叶变换（FFT），网格法，极大似然法，01B工交车调度问题多目标规划02A车灯线光源的优化非线性规划数值模拟，微元法，函数最值，02B彩票问题单目标决策效用函数法，模糊数学中的隶属度函数，层次分析法，分类加权法，熵值法，logistic函数03ASARS的传播微分方程、差分方程负反馈系统，时间序列模型，神经网络，分支过程的MonteCarlo仿真，龙格-库塔法，曲线拟合，smallworldnetwork，sznajd模型，元胞自动机03B露天矿生产的车辆安排整数规划、运输问题贪心算法，04A奥运会临时超市网点设计统计分析、数据处理、优化聚类分析，点阵模型，数据挖掘（apriori 算法）04B电力市场的输电阻塞管理数据拟合、优化huffman决策树，启发式算法05A长江水质的评价和预测预测评价、数据处理逼近理想解排序法，GM（1，1）模型，时间序列分析，反应扩散方程，二元线性回归预测，模糊综合评价法，置信水平，归一化法，主成份分析法，05BDVD在线租赁随机规划、整数规划0-1规划，贪婪算法，最小费用最大流06A出版社书号问题预测评价、数据处理出版社的资源配置06BHiv病毒问题随机规划、整数规划艾滋病疗法的评价07A人口问题整数规划、数据处理、优化人口预测，常微分方程，状态空间分析法07B公交车问题多目标规划、动态规划、图论、0-1规划最短路算法，集合求教算法，08A照相机问题非线性方程组、优化08B大学学费问题数据收集和处理、统计分析、回归分析09A制动器试验台的控制方法分析微元分析法09B眼科病床的合理安排层次分析法整数规划动态规划10A储油罐的变位识别与罐容表标定非线性规划多元拟合10B上海世博会影响力的定量评估数据收集和处理，层次分析法时间序列分析从问题的解决方法上分析，涉及到的数学建模方法：几何理论、组合概率、统计(回归)分析、优化方法（规划）、图论与网络优化、层次分析、插值与拟合、差分方法、微分方程、排队论、模糊数学、随机决策、多目标决策、随机模拟、灰色系统理论、神经网络、时间序列、综合评价、机理分析等方法。

历年美国大学生数学建模竞赛试题MCM.(翻译版)doc

1985 年美国大学生数学建模竞赛MCM 试题1985年MCM:动物种群选择合适的鱼类和哺乳动物数据准确模型。

模型动物的自然表达人口水平与环境相互作用的不同群体的环境的重要参数,然后调整账户获取表单模型符合实际的动物提取的方法。

包括任何食物或限制以外的空间限制,得到数据的支持。

考虑所涉及的各种数量的价值,收获数量和人口规模本身,为了设计一个数字量代表的整体价值收获。

找到一个收集政策的人口规模和时间优化的价值收获在很长一段时间。

检查政策优化价值在现实的环境条件。

1985年MCM B:战略储备管理钴、不产生在美国,许多行业至关重要。

(国防占17%的钴生产。

1979年)钴大部分来自非洲中部,一个政治上不稳定的地区。

1946年的战略和关键材料储备法案需要钴储备,将美国政府通过一项为期三年的战争。

建立了库存在1950年代,出售大部分在1970年代初,然后决定在1970年代末建立起来,与8540万磅。

大约一半的库存目标的储备已经在1982年收购了。

建立一个数学模型来管理储备的战略金属钴。

你需要考虑这样的问题:库存应该有多大?以什么速度应该被收购?一个合理的代价是什么金属?你也要考虑这样的问题:什么时候库存应该画下来吗?以什么速度应该是画下来吗?在金属价格是合理出售什么?它应该如何分配?有用的信息在钴政府计划在2500万年需要2500万磅的钴。

美国大约有1亿磅的钴矿床。

生产变得经济可行当价格达到22美元/磅(如发生在1981年)。

要花四年滚动操作,和thsn六百万英镑每年可以生产。

1980年,120万磅的钴回收,总消费的7%。

1986 年美国大学生数学建模竞赛MCM 试题1986年MCM A:水文数据下表给出了Z的水深度尺表面点的直角坐标X,Y在码(14数据点表省略)。

深度测量在退潮。

你的船有一个五英尺的草案。

你应该避免什么地区内的矩形(75200)X(-50、150)?1986年MCM B:Emergency-Facilities位置迄今为止,力拓的乡牧场没有自己的应急设施。

历年参赛题目

中国最早从1989年有北京地区的清华、北大、北京理工等学校派队参赛，近几年来仅中国的参赛学校及队数几乎都占了参赛总数的三分之一以上，而且每年都能取得最高奖。中国的大学生数学建模竞赛（CUMCM）始于1992年，首先由中国工业与应用数学学会（CSIAM）举办了民间的“全国大学生数学建模竞赛”，到1994年是由原国家教委高教司直接领导组织，由工业与应用数学学会具体承办的一项大规模的竞赛活动。
1994 1995 1996
A B A B A B
逢山开路锁具装箱一个飞行管理问题天车与冶炼炉的作业调度节水洗衣机问题最优捕鱼问题
1997 1998 1999 2000
A B A B A B A B
零件的参数设计最优截断切割问题投资的收益和风险灾情巡视路线自动化车床管理钻井布局 DNA 序列分类钢管订购和运输
五、近几年全国大学生数学建模竞赛题
数学建模竞赛（MCM）最早始于美国，1985年由美国政府部门资助，由美国数学及其应用联合会（COMAP）主办，由美国工业与应用数学学会（SIAM）、运筹及工业和应用数学协会（INFORMS）及数学学会（MAA）协办。第一届MCM只有70所高校90个参赛队，后来它的影响力逐步扩大，现已成为有十几个国家和地区参加的国际型的竞赛活动。
2001 2002 2003 2004
A B A B A B A B
血管的三维重建公交车调度车灯线光源的优化设计彩票中的数学非典型肺炎的传染和控制露天矿生产的车辆安排奥运会临时超市网点设计评价和预测 DVD 在线租赁出版社的资源配置艾滋病疗效的评价及疗效的预测
2006
B
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历年来的MCM题
• • • • • • • • • • 1985年A题：动物群体管理 B题：战略物资储备的管理 1986年A题：水道测量数据 B题：应急设施的位置 1987年A题：盐的存贮 B题：停车场 1988年A题：确定毒品走私船的位置 B题：两辆铁路平板车的装货问题 1989年A题：蠓的分类 B题：飞机排队

数学建模历年竞赛试题

目录前言................................................................................................. 错误!未定义书签。

目录........................................................................................................................... - 0 - 一、什么是数学模型............................................................................................... - 3 -2001年B题……公交车调度......................................................................... - 4 - 2001年C题……基金使用计划..................................................................... - 9 - 2002年A题……车灯线光源的优化设计................................................... - 10 - 2002年B题……彩票中的数学................................................................... - 11 - 2003年A题……SARS的传播.................................................................... - 15 - 2003年B题……露天矿生产的车辆安排................................................... - 26 - 2003年D题……抢渡长江........................................................................... - 29 - 2004年C题……饮酒驾车........................................................................... - 32 - 2004年B题……电力市场的输电阻塞管理............................................... - 34 - 电力市场交易规则：............................................................................. - 35 -输电阻塞管理原则：............................................................................. - 36 -表1各机组出力方案（单位：兆瓦，记作MW） ............................ - 39 -表2各线路的潮流值（各方案与表1相对应，单位：MW） ......... - 41 -表3各机组的段容量（单位：MW） ................................................. - 42 -表4各机组的段价（单位：元/兆瓦小时，记作元/MWh）............. - 42 -表5各机组的爬坡速率（单位：MW/分钟） .................................... - 43 -表6各线路的潮流限值（单位：MW）和相对安全裕度 ................. - 43 -2008年B题……高等教育学费标准探讨................................................... - 43 - 2008年D题……NBA赛程的分析与评价 ................................................. - 45 - 2009年A题……制动器试验台的控制方法分析....................................... - 47 - 2009年B题……眼科病床的合理安排....................................................... - 50 - 【附录】2008-07-13到2008-09-11的病人信息 ................................ - 51 - 2009年D题……会议筹备........................................................................... - 77 - 附表1……10家备选宾馆的有关数据................................................. - 78 -附表2……本届会议的代表回执中有关住房要求的信息（单位：人）- 79 -附表3……以往几届会议代表回执和与会情况.................................. - 80 -附图（其中500等数字是两宾馆间距，单位为米）......................... - 81 -二、为什么要学习数学模型................................................................................. - 83 -1、数学模型无处不在，我们的生活、工作、学习都离不开它............... - 83 -例1买房贷款问题................................................................................. - 83 -例2物体冷却过程的数学模型............................................................. - 84 -2、是学好数学用好数学的必经之路........................................................... - 86 -3、是数学教学改革的重要手段和有效路径............................................... - 88 -4、数学建模竞赛所提唱的团队精神是现代大学生必须具备素质........... - 91 -5、数学建模竞赛鼓励学生用跳跃式的、发散式的形象思维方法，这有利于培养学生的创新意识。

1988B 数学建模MCM题目

Other Problems of the First Ten Years233 1988:The Railroad Flatcar Problem Two railroadﬂatcars 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).Table1gives,for each crate,the thickness,weight,and number available.Each car has10.2meters of length available for packing the crates(like slices of toast)and can carry up to40metric tons.There is a special constraint on the total number of C5,C6,and C7crates because of a subsequent local trucking restriction:The total space(thickness)occupied by these crates must not exceed302.7cm.Load the twoﬂatcars(see Figure1 so as to minimize the wastedﬂoor space.Table1.The thickness,weight,and number of each kind of crate.C1C2C3C4C5C6C7t48.752.061.372.048.752.064.0cmw2,0003,0001,0005004,0002,0001,000kg8796648Figure1.Diagram of loading of aﬂatcar.234Tools for Teaching1994Comments by the Contest DirectorThe problem was suggested by John J.Bartholdi III(School of Industrial and Systems Engineering,Georgia Institute of Technology).It is a modiﬁ-cation of an unsolved problem that surfaced at the Ford Motor Company.The Outstanding papers were by teams from Harvard University,Uni-versity of California–Berkeley,University of Toronto,and the itary Academy.Their papers,together with a commentary by problem author Bartholdi,were published in The UMAP Journal9(4)(1988):343–403.。

数学建模课程及答案

《数学建模课程》练习题一一、填空题1. 设开始时的人口数为0x ，时刻t 的人口数为)(t x ，若人口增长率是常数r ，那麽人口增长问题的马尔萨斯模型应为。

2. 设某种商品的需求量函数是,1200)(25)(+-=t p t Q 而供给量函数是3600)1(35)(--=t p t G ，其中)(t p 为该商品的价格函数，那麽该商品的均衡价格是。

3. 某服装店经营的某种服装平均每天卖出110件，进货一次的手续费为200元，存储费用为每件0.01元/天，店主不希望出现缺货现象，则最优进货周期与最优进货量分别为。

4. 一个连通图能够一笔画出的充分必要条件是 .5.设开始时的人口数为0x ，时刻t 的人口数为)(t x ，若允许的最大人口数为m x ，人口增长率由sx r x r -=)(表示，则人口增长问题的罗捷斯蒂克模型为 .6. 在夏季博览会上，商人预测每天冰淇淋销量N 将和下列因素有关：（1）参加展览会的人数n ；（2）气温T 超过C10；（3）冰淇淋的售价p .由此建立的冰淇淋销量的比例模型应为 .7、若银行的年利率是x %，则需要时间，存入的钱才可翻番. 若每个小长方形街路的8. 如图是一个邮路，邮递员从邮局A 出发走遍所有长方形街路后再返回邮局. 边长横向均为1km ，纵向均为2km ，则他至少要走 km.. A9. 设某种新产品的社会需求量为无限，开始时的生产量为100件，且设产品生产的增长率控制在0.1，t 时刻产品量为)(t x ，则)(t x = .10. 商店以10元/件的进价购进衬衫，若衬衫的需求量模型是802,Q p p =-是销售单价（元/件），为获得最大利润，商店的出售价是 .二、分析判断题1．从下面不太明确的叙述中确定要研究的问题，需要哪些数据资料（至少列举3个），要做些甚麽建模的具体的前期工作（至少列举3个），建立何种数学模型：一座高层办公楼有四部电梯，早晨上班时间非常拥挤，该如何解决。

2000-2013美国数学建模竞赛(MCM、ICM)历年试题汇总(1)

目录2000 年美国大学生数学建模竞赛MCM、ICM 试题 (3)2000 MCM A: Air Traffic Control (3)2000 MCM B: Radio Channel Assignments (3)2000 ICM: Elephants: When is Enough, Enough? (5)2001 年美国大学生数学建模竞赛MCM、ICM 试题 (7)2001 MCM A: Choosing a Bicycle Wheel (7)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...). (8)2001 ICM: Our Waterways - An Uncertain Future (10)2002 年美国大学生数学建模竞赛MCM、ICM 试题 (14)2002 MCM A: Wind and Waterspray (14)2002 MCM B: Airline Overbooking (14)2002 ICM: Scrub Lizards (15)2003 年美国大学生数学建模竞赛MCM、ICM 试题 (19)2003 MCM A: The Stunt Person (19)2003 MCM B: Gamma Knife Treatment Planning (19)2003 ICM: Aviation Baggage Screening Strategies: To Screen or Not to Screen, that is the Question (20)2004 年美国大学生数学建模竞赛MCM、ICM 试题 (24)2004 MCM A: Are Fingerprints Unique? (24)2004 MCM B: A Faster QuickPass System (24)2004 ICM: To Be Secure or Not to Be? (24)2005 年美国大学生数学建模竞赛MCM、ICM 试题 (25)2005 MCM A: Flood Planning (25)2005 MCM B: Tollbooths (25)2005 ICM: Nonrenewable Resources (25)2006 年美国大学生数学建模竞赛MCM、ICM 试题 (27)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation (27)2006 MCM B: Wheel Chair Access at Airports (27)2006 ICM: Trade-offs in the fight against HIV/AIDS (28)2007 年美国大学生数学建模竞赛MCM、ICM 试题 (32)2007 MCM A: Gerrymandering (32)2007 MCM B: The Airplane Seating Problem (32)2007 ICM: Organ Transplant: The Kidney Exchange Problem (33)2008 年美国大学生数学建模竞赛MCM、ICM 试题 (38)2008 MCM A: Take a Bath (38)2008 MCM B: Creating Sudoku Puzzles (38)2008 ICM: Finding the Good in Health Care Systems (38)2009 年美国大学生数学建模竞赛MCM、ICM 试题 (40)2009 MCM A: Designing a Traffic Circle (40)2009 MCM B: Energy and the Cell Phone (40)2009 ICM: Creating Food Systems: Re-Balancing Human-Influenced Ecosystems41 2010年美国大学生数学建模竞赛 MCM、ICM 试题 (42)2010 MCM A: The Sweet Spot (42)2010 MCM B: Criminology (43)2010 ICM: The Great Pacific Ocean Garbage Patch (44)2011年美国大学生数学建模竞赛 MCM、ICM 试题 (45)2011 MCM A: Snowboard Course (45)2011 MCM B: Repeater Coordination (45)2011 ICM: Environmentally and Economically Sound (46)2012年美国大学生数学建模竞赛 MCM、ICM 试题 (48)2012 MCM A: The Leaves of a Tree (48)2012 MCM B: Camping along the Big Long River (50)2012 ICM: Modeling for Crime Busting (51)2013年美国大学生数学建模竞赛 MCM、ICM 试题 (59)2013 MCM A: The Ultimate Brownie Pan (59)2013 MCM B: Water, Water, Everywhere (61)2013 ICM: NetworkModeling of Earth's Health (62)2000 年美国大学生数学建模竞赛MCM、ICM 试题2000 MCM A: Air Traffic ControlTo improve safety and reduce air traffic controller workload, the Federal Aviation Agency (FAA) is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analysit at the FAA has posed the following problems.Requirement A: Given two airplanes flying in space, when should the air traffic controller consider the objects to be too close and to require intervention? Requirement B: And airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how do we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of aircraft simultaneously passing through that sector1.at any one instant?2.during any given interval of time?3.during a particular time of day?How does the number of potential conflicts arising during those periods affect complexity? Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this complexity? In addition to the guidelines for your report, write a summary (no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusions.2000 MCM B: Radio Channel AssignmentsWe seek to model the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grix (honeycomb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon.An interval of the frequency spectrum is to be alloted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1,2,3, … . Each transmitter wil be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided.Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assugn channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span.Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interference.Requirement A: There are several contrainsts on the frequency assignments. First, no two transmitters within distance 4s of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these contraints, what can we say about the span in Figure 1?Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions.Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance 2s differ by at least some given integer k, while those at distance at most 4s must still differ by at least one. What cna we say about the span and about efficient strategies for designing assignments, as a function of k?Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider?Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findings.2000 ICM: Elephants: When is Enough, Enough?“Ultimately, if a habitat is undesirably changed by elephants, then their removal should be considered -even by culling.”National Geographic (Earth Almanac) –December 1999 A large National Park in South Africa contains approximately 11,000 elephants. Management policy requires a healthy environment that can maintain a stable herf of 11,000 elephants. Each year park rangers count the elephant population. During the past 20 years whole herds have been removed to keep the population as close to 11,000 as possible. The process involved shooting (for the most part) and occasionally relocating approximately 600 to 800 elephants per year.Recently, there has been a public outcry against the shooting of these elephants. In addition, it is no longer feasible to relocate even a small population of elephants each year. A contraceptive dart, however, has been developed that can prevent a mature elephant cow from conceiving for a period of two years.Here is some information about eh elephants in the Park:∙There is very little emigration of immigration of elephants.∙The gender ratio is very close to 1:1 and control measures have endeavored to maintain parity.∙The gender ratio of newborn calves is also about 1:1. Twins are bornaverage, a calf every 3.5 years until they reach an age of about 60.Gestation is approximately 22 months.∙The contraceptive dart causes an elephant cow to come into oestrus every month (but not conceiving). Elephants usually have courtship only once in 3.5 years, so the monthly cycle can cause additional stress.∙ A cow can be darted every year without additional detrimental effects. A mature elephant cow will not be able to conceive for 2 years after thelast darting.∙Between 70% and 80% of newborn calves survive to age 1 year.Thereafter, the survival rate is uniform across all ages and is very high(over 95%), until about age 60; it is a good assumption that elephantsdie before reading age 70.There is no hunting and negligible poaching in the Park.The park management has a rough data file of the approximate ages and2 years. This data is available on website: icm2000data.xls. Unfortunately no data is available for the elephants that have been shot or remain in the Park.Your overall task is to develop and use models to investigate how the contraceptive dart might be used for population control. Specifically:Task 1: Develop and use a model to speculate about the likely survival rate for elephants aged 2 to 60. Also speculate about the current age structure of the elephant population.Task 2: Estimate how many cows would need to be darted each year to keep the population fixed at approximately 11,000 elephants. Show how the uncertainty in the data at your disposal affects your estimate. Comment on any changes in the age structure of the population and how this might affect tourists. (You may want to look ahead about 30-60 years.)Task 3: If it were feasible to relocate between 50 and 300 elephants per year, how would this reduce the number of elephants to be darted? Comment on the trade-off between darting and relocation.large number of elephants (due to disease or uncontrolled poaching), even if darting stopped immediately, the ability of the population to grow again would be seriously impeded. Investigate and respond to this concer.Task 5: The management in the Park is skeptical about modeling. In particular, they argue that a lack of complete data makes a mockery of any attempt to use models to guide their decision. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the park management that responds to their concerns and provides advice. Also suggest ways to increase the park managers confidence in your model and your conclusions.Task 6: If your model works, other elephant parks in Africa would be interested in using it. Prepare a darting plan for parks of various sizes (300-25,000 elephants), with slightly different survival rates and transportation possibilities.2001 年美国大学生数学建模竞赛MCM、ICM 试题2001 MCM A: Choosing a Bicycle WheelCyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike.Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course.Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right.The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear.Task 1. Provide a table giving the wind speed at which the power required for a solid rear wheel is less than for a spoked rear wheel. The table should include the wind speeds for different road grades startingfrom zero percent to ten percent in one percent increments. (Roadgrade is defined to be the ratio of the total rise of a hill divided by thelength of the road. If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportionalto the road grade. A rider will lose about 8 kph for a five percent grade over 100 meters.∙Task 2. Provide an example of how the table could be used for a specific time trial course.∙Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision.2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...)Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare.The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned.A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of the coastal strip. (More accurate data, if sought, are widely available.)The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees (including some coming from farther north by other routes), so some traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only very slowly. Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed:1.Under what conditions does the plan for turning the two coastal-boundlanes of I-26 into two lanes of Columbia-bound traffic, essentiallyturning the entire I-26 into one-way traffic, significantly improveevacuation traffic flow?2.In 1999, the simultaneous evacuation of the state's entire coastal regionwas ordered. Would the evacuation traffic flow improve under analternative strategy that staggers the evacuation, perhapscounty-by-county over some time period consistent with the pattern of how hurricanes affect the coast?3.Several smaller highways besides I-26 extend inland from the coast.Under what conditions would it improve evacuation flow to turn around traffic on these?4.What effect would it have on evacuation flow to establish moretemporary shelters in Columbia, to reduce the traffic leaving Columbia?5.In 1999, many families leaving the coast brought along their boats,campers, and motor homes. Many drove all of their cars. Under whatconditions should there be restrictions on vehicle types or numbers ofvehicles brought in order to guarantee timely evacuation?6.It has been suggested that in 1999 some of the coastal residents ofGeorgia and Florida, who were fleeing the earlier predicted landfalls ofHurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation trafficflow? Clearly identify what measures of performance are used tocompare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.Clearly identify what measures of performance are used to compare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.2001 ICM: Our Waterways - An Uncertain FutureZebra mussels, Dreissena polymorpha, are small, fingernail-sized, freshwater mollusks unintentionally introduced to North America via ballast water from a transoceanic vessel. Since their introduction in the mid 1980s, they have spread through all of the Great Lakes and to an increasing number of inland waterways in the United States and Canada. Zebra mussels colonize on various surfaces,such as docks, boat hulls, commercial fishing nets, water intake pipes and valves, native mollusks and other zebra mussels. Their only known predators, some diving ducks, freshwater drum, carp, and sturgeon, are not numerous enough to have a significant effect on them. Zebra mussels have significantly impacted the Great Lakes ecosystem and economy. Many communities are trying to control or eliminate these aquatic pests. SOURCE: Great Lakes Sea Grant Network /.Researchers are attempting to identify the environmental variables related to the zebra mussel infestation in North American waterways. The relevant factors that may limit or prevent the spread of the zebra mussel are uncertain. You will have access to some reference data to include listings of several chemicals and substances in the water system that may affect the spread of the zebra mussel throughout waterways. Additionally, you can assume individual zebra mussels grow at a rate of 15 millimeters per year with a life span between 4 - 6 years. The typical mussel can filter 1 liter of water each day.Requirement A: Discuss environmental factors that could influence the spread of zebra mussels.Requirement B: Utilizing the chemical data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAChem1.xls, and the mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 1.xls model the population growth of zebra mussels in Lake A. Be sure to review the Information about the collection of the zebra mussel data. Requirement C: Utilizing additional data on Lake A from another scientist provided at :ap/undergraduate/contests/icm/imagesdata/LakeAChem2.xls and additional mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 2.xls corroborate the reasonableness of your model from Requirement B. As a result of this additional data, adjust your earlier model. Analyze the performance of your model. Discuss the sensitivity of your model. Requirement D: Utilizing the Chemical data from two lakes (Lake B and Lake C) in the United States provided atap/undergraduate/contests/icm/imagesdata/LakeB.xls and ap/undergraduate/contests/icm/imagesdata/LakeC.xls determine if these lakes are vulnerable to the spread of zebra mussels. Discuss your prediction.Requirement E: The community in the vicinity of Lake B (in requirement D) is considering specific policies for the de-icing of roadways near the lake duringthe winter season. Provide guidance to the local government officials regarding a policy on “de-icing agents.”In your guidance include predictions on the long-term impact of de-icing on the zebra mussel population. Requirement F: It has been recommended by a local community in the United States to introduce round goby fish. Zebra mussels are not often eaten by native fish species so they represent a dead end ecologically. However, round gobies greater than 100 mm feed almost exclusively on zebra mussels. Ironically, because of habitat destruction, the goby is endangered in its native habitat of the Black and Caspian Seas in Russia. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the local community leaders that responds to their recommendation to introduce the round goby. Also suggest ways to help reduce the growth of the mussel within and among waterways.Information about the collection of the zebra mussel dataThe developmental state of the Zebra mussel is categorized by three stages: veligers (larvae), settling juveniles, and adults. Veligers (microscopic zebra mussel larvae) are free-swimming, suspended in the water for one to three weeks, after which they begin searching for a hard surface to attach to and begin their adult life. Looking for zebra mussel veligers is difficult because they are not easily visible by the naked eye. Settled juvenile zebra mussels can be felt on smooth surfaces like boats and motors. An advanced zebra mussel infestation can cover a surface, even forming thick mats sometimes reaching very high densities. The density of juveniles was determined along the lake using three 15×15 cm settling plates. The top plate remained in the water for the entire sampling season (S - seasonal) to estimate seasonal accumulation. The middle and bottom plates are collected after specific periods (A –alternating ) of time denoted by “Lake Days”in the data files.The settling plates are placed under the microscope and all juveniles on the undersides of the plate are counted and densities are reported as juveniles/m^2.2002 年美国大学生数学建模竞赛MCM、ICM 试题2002 MCM A: Wind and WatersprayAn ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area. Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.2002 MCM B: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations. Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.Consider the overbooking issue in light of the current situation: Less flights by airlines from point A to point B Heightened security at and around airports Passengers' fear Loss of billions of dollars in revenue by airlines to dateBuild a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company's revenue is maximized. Insure that your model reflects the issues above, andconsider alternatives for handling “bumped”passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.2002 ICM: Scrub LizardsThe Florida scrub lizard is a small, gray or gray-brown lizard that lives throughout upland sandy areas in the Central and Atlantic coast regions of Florida. The Florida Committee on Rare and Endangered Plants classified the scrub lizard as endangered.You will find a fact sheet on the Florida Scrub Lizard at/undergraduate/contests/mcm/contests/2002/problem s/icm2002data/scrublizard.pdfThe long-term survival of the Florida scrub lizard is dependent upon preservation of the proper spatial configuration and size of scrub habitat patches.Task 1: Discuss factors that may contribute to the loss of appropriate habitat for scrub lizards in Florida. What recommendations would you make to the state of Florida to preserve these habitats and discuss obstacles to the implementation of your recommendations?average fecundity of adult lizards); Sj (the survivorship of juvenile lizards- between birth and the first reproductive season); and Sa (the average adult survivorship).Table 1Summary data for a cohort of scrub lizards captured and followed for 4 consecutive years. Hatchling lizards (age 0) do not produce eggs during the summer they are born. Average clutch size for all other females is proportional to body size according to the function y = 0.21*(SVL)-7.5, where y is the clutch size and SVL is the snout-to-vent length in mm.Year Age Total NumberLivingNumber of LivingFemalesAvg. Female Size(mm)1 0 972 495 30.32 1 180 92 45.83 2 20 11 55.84 3 2 2 56.0Task 3: It has been conjectured that the parameters Fa , Sj , and Sa , are related to the size and amount of open sandy area of a scrub patch. Utilize the data provided in Table 2 to develop functions that estimate Fa, Sj , and Sa for different patches. In addition, develop a function that estimates C, the carrying capacity of scrub lizards for a given patch.Table 2Summary data for 8 scrub patches including vital rate data for scrub lizards. Annual female fecundity (Fa), juvenile survivorship (Sj), and adult survivorship (Sa) are presented for each patch along with patch size and the amount of open sandy habitat.Patch Patch Size (ha) Sandy Habitat (ha) Fa Sj Sa Density (lizards/ha)a 11.31 4.80 5.6 0.12 0.06 58b 35.54 11.31 6.6 0.16 0.10 60c 141.76 51.55 9.5 0.17 0.13 75d 14.65 7.55 4.8 0.15 0.09 55e 63.24 20.12 9.7 0.17 0.11 80f 132.35 54.14 9.9 0.18 0.14 82g 8.46 1.67 5.5 0.11 0.05 40h 278.26 84.32 11.0 0.19 0.15 115Task 4: There are many animal studies that indicate that food, space, shelter, or even reproductive partners may be limited within a habitat patch causing individuals to migrate between patches. There is no conclusive evidence on why scrub lizards migrate. However, about 10 percent of juvenile lizards do migrate between patches and this immigration can influence the size of the population within a patch. Adult lizards apparently do not migrate. Utilizing the data provided in the histogram below estimate the probability of lizards surviving the migration between any two patches i and patch j.Table 3HistogramMigration data for juvenile lizards marked, released, and recaptured up to 6 months later. Surveys for recapture were conducted up to 750m from release sites.Task 5: Develop a model to estimate the overall population size of scrub lizards for the landscape given in Table 3. Also, determine which patches are suitable for occupation by scrub lizards and which patches would not support a viable population.Patch size and amount of open sandy habitat for a landscape of 29 patches located on the Avon Park Air Force Range. See:/undergraduate/contests/icm/2002problem/map.jpg for a map of the landscape.Patch Identification Patch Size (ha) Sandy Habitat (ha)1 13.66 5.382 32.74 11.913 1.39 0.234 2.28 0.765 7.03 3.626 14.47 4.387 2.52 1.998 5.87 2.499 22.27 8.44。

1988年美国大学生数学建模竞赛试题

MCM1988A 铁路平板车问题7种规格的包装箱要装有两辆铁路平板车上去，包装箱的宽和高相同，但厚度（t,以cm计）和重量（,ω以kg计）不同，表A-1给出了每包装箱的厚度、重量和数量，每辆车有10.2m长的地方用来装包装箱（像面包片那样），车的载重为40吨，对C5、C6、C7、规格的包装箱的总数有一个特殊的限制：这些规格箱子所占的空间（厚度）不能超过302.7cm。

试把包装箱装到两辆平板车上去（图A-6）使得浪费的空间最小。

表A-1 每种包装箱的厚度、重量和数量C 1C2C3C4C5C6C7t ω48.72000852.03000761.31000972.0500648.74000652.02000464.010008cmkg MCM1988B 毒品走私船的问题相距5.43英里的两个监听站收到一个短暂的无线电信号，收听到信号时两台测向仪分别定位于110°和119°（图A-7），测向仪的精度在2°以内，该信号来自一个毒品交易活跃的区域，据推测一只机动船正等着有人来取毒品，当时正值黄昏，风平浪静，没有潮流，一架小型直升飞机离开监听站1的升降台，并能精确地沿着110°的方向飞行，直升飞机的飞行速度是船的3倍，在离船500英尺时，船上能听到直升飞机的声音，直升飞机只有一种侦察装置——探照灯，在200英尺的地方探照灯只能照亮半径为25英尺的圆形区域。

·说明飞行员能找到正等着的毒品船的（最小）区域。

·研究一种直升飞机的最佳搜索方法。

你的计算中要有95%的置信度。

数学建模基础(入门必备)

一、数学模型的定义现在数学模型还没有一个统一的准确的定义，因为站在不同的角度可以有不同的定义。

不过我们可以给出如下定义：“数学模型是关于部分现实世界和为一种特殊目的而作的一个抽象的、简化的结构。

”具体来说，数学模型就是为了某种目的，用字母、数学及其它数学符号建立起来的等式或不等式以及图表、图象、框图等描述客观事物的特征及其内在联系的数学结构表达式。

一般来说数学建模过程可用如下框图来表明：数学是在实际应用的需求中产生的，要解决实际问题就必需建立数学模型，从此意义上讲数学建模和数学一样有古老历史。

例如，欧几里德几何就是一个古老的数学模型，牛顿万有引力定律也是数学建模的一个光辉典范。

今天，数学以空前的广度和深度向其它科学技术领域渗透，过去很少应用数学的领域现在迅速走向定量化，数量化，需建立大量的数学模型。

特别是新技术、新工艺蓬勃兴起，计算机的普及和广泛应用，数学在许多高新技术上起着十分关键的作用。

因此数学建模被时代赋予更为重要的意义。

二、建立数学模型的方法和步骤1. 模型准备要了解问题的实际背景，明确建模目的，搜集必需的各种信息，尽量弄清对象的特征。

2. 模型假设根据对象的特征和建模目的，对问题进行必要的、合理的简化，用精确的语言作出假设，是建模至关重要的一步。

如果对问题的所有因素一概考虑，无疑是一种有勇气但方法欠佳的行为，所以高超的建模者能充分发挥想象力、洞察力和判断力，善于辨别主次，而且为了使处理方法简单，应尽量使问题线性化、均匀化。

3. 模型构成根据所作的假设分析对象的因果关系，利用对象的内在规律和适当的数学工具，构造各个量间的等式关系或其它数学结构。

这时，我们便会进入一个广阔的应用数学天地，这里在高数、概率老人的膝下，有许多可爱的孩子们，他们是图论、排队论、线性规划、对策论等许多许多，真是泱泱大国，别有洞天。

不过我们应当牢记，建立数学模型是为了让更多的人明了并能加以应用，因此工具愈简单愈有价值。

历年全国赛数学建模题目

目录1996年全国大学生数学建模竞赛题目 (2)A题最优捕鱼策略 (2)B题节水洗衣机 (2)1997年全国大学生数学建模竞赛题目 (3)A题零件的参数设计 (3)B题截断切割 (4)1998年全国大学生数学建模竞赛题目 (5)A题投资的收益和风险 (5)B题灾情巡视路线 (6)1999创维杯全国大学生数学建模竞赛题目 (7)A题自动化车床管理 (7)B题钻井布局 (8)C题煤矸石堆积 (9)D题钻井布局（同 B 题） (9)2000网易杯全国大学生数学建模竞赛题目 (10)A题 DNA分子排序 (10)B题钢管订购和运输 (12)C题飞越北极 (15)D题空洞探测 (15)2001年全国大学生数学建模竞赛题目 (17)A题血管的三维重建 (17)B题公交车调度 (18)C题基金使用计划 (20)D题公交车调度 (20)2002高教社杯全国大学生数学建模竞赛题目 (21)A题车灯线光源的优化设计 (21)B题彩票中的数学 (21)C题车灯线光源的计算 (23)D题赛程安排 (23)2003高教社杯全国大学生数学建模竞赛题目 (24)A题 SARS的传播 (24)B题露天矿生产的车辆安排 (28)C题 SARS的传播 (29)D题抢渡长江 (30)2004高教社杯全国大学生数学建模竞赛题目 (31)A题奥运会临时超市网点设计 (31)B题电力市场的输电阻塞管理 (35)C题饮酒驾车 (39)D题公务员招聘 (39)2005高教社杯全国大学生数学建模竞赛题目 (42)A题: 长江水质的评价和预测 (42)B题： DVD在线租赁 (43)C题雨量预报方法的评价 (44)D题： DVD在线租赁 (45)2006高教社杯全国大学生数学建模竞赛题目 (46)A题: 出版社的资源配置 (46)B题: 艾滋病疗法的评价及疗效的预测 (46)C题: 易拉罐形状和尺寸的最优设计 (47)D题: 煤矿瓦斯和煤尘的监测与控制 (48)2007高教社杯全国大学生数学建模竞赛题目 (53)A题：中国人口增长预测 (53)2008高教社杯全国大学生数学建模竞赛题目 (56)A题数码相机定位 (56)B题高等教育学费标准探讨 (57)C题地面搜索 (57)2009高教社杯全国大学生数学建模竞赛题目 (59)A题制动器试验台的控制方法分析 (59)B题眼科病床的合理安排 (60)C题卫星和飞船的跟踪测控 (61)D题会议筹备 (61)2010全国高教社杯数学建模题目 (65)A题储油罐的变位识别与罐容表标定 (65)B题 2010年上海世博会影响力的定量评估 (66)A题最优捕鱼策略为了保护人类赖以生存的自然环境,可再生资源(如渔业、林业资源)的开发必须适度.一种合理、简化的策略是,在实现可持续收获的前提下,追求最大产量或最佳效益.考虑对某种鱼(鳀鱼)的最优捕捞策略:假设这种鱼分四个年龄组,称1龄鱼,…,4龄鱼,各年龄组每条鱼的平均重量分别为 5.07,11.55,17.86,22.99(g),各年龄组鱼的自然死亡率为0.8(1/年),这种鱼为季节性集产卵繁殖,平均每条4龄鱼的产卵量为1.109× (个),3龄鱼的产卵量为这个数的一半,2龄鱼和1龄鱼不产卵,产卵和孵化期为每年的最后4个月,卵孵化并成活为1龄鱼,成活率(1龄鱼条数与产卵总量n之比)为1.22× /(1.22× +n).渔业管理部门规定,每年只允许在产卵孵化期前的8个月内进行捕捞作业.如果每年投入的捕捞能力(如渔船数﹑下网次数等)固定不变,这时单位时间捕捞量与各年龄组鱼群条数成正比,比例系数不妨称捕捞强度系数.通常使用13mm网眼的拉网,这种网只能捕3龄鱼和4龄鱼,其两个捕捞强度系数之比为0.42:1.渔业上称这种方式为固定努力量捕捞.1)建立数学模型分析如何实现可持续捕获(即每年开始捕捞时鱼场中各年龄组鱼群不变),并且在此前提下得到最高的年收获量(捕捞总重量).2)某渔业公司承包这种鱼的捕捞业务5年,合同要求5年后鱼群的生产能力不能受到太大破坏. 已知承包时各年龄组鱼群的数量分别为:122,29.7,10.1,3.29(×条),如果任用固定努力量的捕捞方式,该公司应采取怎样的策略才能使总收获量最高.(北京师范大学刘来福提供)B题节水洗衣机我国淡水资源有限,节约用水人人又责,洗衣在家庭用水中占有相当大的份额,目前洗衣机已相当普及,节约洗衣机用水十分重要.假设在放入衣服和洗涤剂后洗衣机的运行过程为:加水-漂水-脱水-加水-漂洗-脱水-…-加水-漂洗-脱水(称"加水-漂洗-脱水"为运行一轮).请为洗衣机设计一种程序(包括运行多少轮﹑每轮加水量等),使得在满足一定洗涤效果的条件下,总用水量最少.选用合理的数据进行计算,对照目前常用的洗衣机的运行情况,对你的模型和结果做出评价.A题零件的参数设计一件产品由若干零件组装而成，标志产品性能的某个参数取决于这些零件的参数。

1987B停车场问题-赵轶星——大学生数学建模竞赛

1987：停车场问题在新英格兰镇有一个位于街角处、面积100×200平方英尺停车场，场主请你设计它的布局，即设场地上的线怎么画的方案。

容易理解，如果将汽车按照与停车线构成直角的方向，一辆紧挨一辆地排列成行，则可以在停车场内塞进最大数量的汽车。

但是对于缺乏经验的司机来说，按照这种方式停靠车辆是有困难的，它可能造成昂贵的保险费用支出。

为了减少因停车造成意外损失的可能性，场主可能不得不雇用一些技术熟练的司机专门停车。

另一方面，如果从通道进入停车位有一个足够大的转弯半径，那么，看来大多数的司机都可以毫无困难地一次停车到位。

当然，通道越宽，场内所能容纳的车辆数目也就越少，这将使得场主减少收入。

1解决一个新英格兰镇的停车场问题摘要给定一个100×200平方英尺的场地，我们分析的目标是确定一种停车场空间的配置方案使得从停车场获得的收入最大。

我们需要考虑专职停车和自助停车两种方案。

自助停车是更好的选择，但是需要一个服务员来收停车费。

为了求得停车场最大的停车数量，我们测试了停车空间的角度从45°到90°的情况。

在如果通道的转角的数量越少，能获得越多空间的假设下，停车场配置的预算的数量可以缩减为7,。

首先，我们分析不考虑入口，出口和服务员的情况。

我们找到了一种能容下76个停车位的配置。

当入口和收费所都考虑进来时，我们的配置相对其他的配置能有更多的停车位（75）。

在我们最后的方案中，移动的许可、下雪时暂时的布置和灯柱的空间都被考虑了。

问题重述1给定一个在新英格兰镇的100×200平方英尺的在转角处的停车场，设计一种配置方案使得停车位最大并且在车场里驾驶的难度最小。

基本假设1、车是自助停车的。

雇佣一个技术熟练的司机来停车的花费对于增加的停车位而增加的收入来说多太多了。

2、停车场里的路都是单向的，这是为了减少通道的宽度和路的总面积。

在这种方法下，停车场的空间可以最大化。

3、入口和出口的位置决定于停车场处于街道转角的位置。

历届美国数学建模竞赛赛题

历届美国数学建模竞赛赛题, 1985－2006AMCM1985问题-A 动物群体的管理AMCM1985问题-B 战购物资储备的管理AMCM1986问题-A 水道测量数据AMCM1986问题-B 应急设施的位置AMCM1987问题-A 盐的存贮AMCM1987问题-B 停车场AMCM1988问题-A 确定毒品走私船的位置AMCM1988问题-B 两辆铁路平板车的装货问题AMCM1989问题-A 蠓的分类AMCM1989问题-B 飞机排队AMCM1990问题-A 药物在脑内的分布AMCM1990问题-B 扫雪问题AMCM1991问题-A 估计水塔的水流量AMCM1992问题-A 空中交通控制雷达的功率问题AMCM1992问题-B 应急电力修复系统的修复计划AMCM1993问题-A 加速餐厅剩菜堆肥的生成AMCM1993问题-B 倒煤台的操作方案AMCM1994问题-A 住宅的保温AMCM1994问题-B 计算机网络的最短传输时间AMCM1995问题-A 单一螺旋线AMCM1995问题-B A1uacha Balaclava学院AMCM1996问题-A 噪音场中潜艇的探测AMCM1996问题-B 竞赛评判问题AMCM1997问题-A Velociraptor(疾走龙属)问题AMCM1997问题-B为取得富有成果的讨论怎样搭配与会成员AMCM1998问题-A 磁共振成像扫描仪AMCM1998问题-B 成绩给分的通胀AMCM1999问题-A 大碰撞AMCM1999问题-B “非法”聚会AMCM1999问题- C 大地污染AMCM2000问题-A空间交通管制AMCM2000问题-B: 无线电信道分配AMCM2000问题-C：大象群落的兴衰AMCM2001问题- A: 选择自行车车轮AMCM2001问题-B：逃避飓风怒吼（一场恶风…）AMCM2001问题-C我们的水系-不确定的前景AMCM2002问题-A风和喷水池AMCM2002问题-B航空公司超员订票AMCM2002问题-C蜥蜴问题AMCM2003问题-A: 特技演员AMCM2003问题-C航空行李的扫描对策AMCM2004问题-A：指纹是独一无二的吗？AMCM2004问题-B：更快的快通系统AMCM2004问题-C安全与否？AMCM2005问题-A：.水灾计划AMCM2005问题-B：TollboothsAMCM2005问题-C：.Nonrenewable ResourcesAMCM2006问题-A：用于灌溉的自动洒水器的安置和移动调度AMCM2006问题-B：通过机场的轮椅AMCM2006问题-C：在与HIV/爱滋病的战斗中的交易AMCM85问题-A 动物群体的管理在一个资源有限，即有限的食物、空间、水等等的环境里发现天然存在的动物群体。

2023初中 midmcm题目

尊敬的读者，今天我要跟大家共享的是2023年初中midmcm题目。

midmcm是中学生数学建模竞赛（Mathematical Contest in Modeling，简称MCM）的一部分，旨在选拔和培养中学生的数学建模能力。

每年的midmcm题目都是围绕着现实生活中的问题展开，要求参赛者进行深入的分析和解决方案的提出。

在2023年的midmcm题目中，主题围绕着“城市交通拥堵”展开。

这是一个备受关注的社会问题，城市交通拥堵给人们的出行带来了诸多不便，也影响着城市的经济和环境。

对于如何有效缓解城市交通拥堵问题，提高城市交通运行效率，是一个具有挑战性和价值的课题。

在这篇文章中，我将从多个角度对2023年初中midmcm题目进行全面评估，并撰写一篇有价值的文章，以展示我对这一主题的深刻理解和个人观点。

让我们从城市交通拥堵的原因和影响展开分析。

城市交通拥堵的原因可以包括车辆数量过多、道路设计不合理、交通信号不协调等方面。

这些原因导致了交通拥堵的严重性和持续性。

而城市交通拥堵所带来的影响也是多方面的，包括时间成本增加、能源消耗增加、环境污染加剧等。

解决城市交通拥堵问题就显得尤为重要。

接下来，让我们从不同的解决方案角度来思考。

在面对城市交通拥堵问题时，我们可以考虑从交通管理、交通规划、公共交通推广等方面入手。

可以通过智能交通管理系统提高交通信号灯的智能性，提高道路通行效率；可以通过合理规划公共交通线路和站点，鼓励市民更多地使用公共交通工具，减少私家车辆使用等方式来缓解城市交通拥堵问题。

在这个过程中，我们也需要考虑技术的应用和创新。

可以利用大数据分析和人工智能技术来预测交通拥堵的状况，以及提供个性化的出行建议；可以利用新能源汽车和共享出行概念来鼓励绿色出行，减少交通拥堵对环境的影响等。

城市交通拥堵是一个复杂的问题，需要从多个方面进行综合分析和解决方案的提出。

通过这个midmcm题目的研究和解答，不仅可以锻炼参赛者的数学建模能力，也可以引导他们关注社会问题，并在思维和创新上进行提升。

1987B停车场问题-赵轶星——大学生数学建模竞赛

1987：停车场问题在新英格兰镇有一个位于街角处、面积100×200平方英尺停车场，场主请你设计它的布局，即设场地上的线怎么画的方案。

容易理解，如果将汽车按照与停车线构成直角的方向，一辆紧挨一辆地排列成行，则可以在停车场内塞进最大数量的汽车。

但是对于缺乏经验的司机来说，按照这种方式停靠车辆是有困难的，它可能造成昂贵的保险费用支出。

为了减少因停车造成意外损失的可能性，场主可能不得不雇用一些技术熟练的司机专门停车。

另一方面，如果从通道进入停车位有一个足够大的转弯半径，那么，看来大多数的司机都可以毫无困难地一次停车到位。

当然，通道越宽，场内所能容纳的车辆数目也就越少，这将使得场主减少收入。

我们需要考虑专职停车和自助停车两种方案。

自助停车是更好的选择，但是需要一个服务员来收停车费。

为了求得停车场最大的停车数量，我们测试了停车空间的角度从45°到90°的情况。

在如果通道的转角的数量越少，能获得越多空间的假设下，停车场配置的预算的数量可以缩减为7,。

首先，我们分析不考虑入口，出口和服务员的情况。

我们找到了一种能容下76个停车位的配置。

当入口和收费所都考虑进来时，我们的配置相对其他的配置能有更多的停车位（75）。

在我们最后的方案中，移动的许可、下雪时暂时的布置和灯柱的空间都被考虑了。

问题重述1给定一个在新英格兰镇的100×200平方英尺的在转角处的停车场，设计一种配置方案使得停车位最大并且在车场里驾驶的难度最小。

基本假设1、车是自助停车的。

雇佣一个技术熟练的司机来停车的花费对于增加的停车位而增加的收入来说多太多了。

2、停车场里的路都是单向的，这是为了减少通道的宽度和路的总面积。

在这种方法下，停车场的空间可以最大化。

3、入口和出口的位置决定于停车场处于街道转角的位置。

1987全国初中数学联赛

1987全国初中数学联赛一、选择题1. 已知实数a ，b ，c 满足0a b c ++=，8abc =，那么111a b c ++的值(A)是正数 (B)是零 (C)是负数 (D)正、负不能确定答( )2. 在一条直线上已知四个不同的点依次是A ，B ，C ，D ，那么到A ，B ，C ，D 的距离之和最小的点(A)可以是直线A D 上的某一点 (B)只是B 点和C 点(C)只是线段A D 的中点 (D)有无穷多个答( )3. 如图，A B 是半圆的直径，O 是圆心，C D A B ⊥，D E O C ⊥，如果A D ，B D 和C D 的长都是有理数，那么命题甲：O E 的长是有理数.乙：D E 的长是有理数.丙：图中所有字母表示的线段的长都是有理数. (A)只有甲正确 (B)只有乙正确(C)只有甲、乙正确 (D)甲、乙、丙都正确答( )4. 已知方程1x ax =+有一个负根而且没有正根，那么a 的取值范围是(A)1a >- (B)1a =- (C)1a ≥ (D)非上述答案答( )5. 已知四边形A B C D 内有一点E ，连接A E ，BE ，C E ，D E ，将四边形A B C D 分成四个面积相等的三角形，那么命题甲：A B C D 是凸四边形.乙：E 是对角线A C 的中点或对角线B D 的中点.丙：A B C D 是平行四边形.在以上三条中(A)只有甲正确 (B)只有乙正确(C)甲、乙、丙都正确 (D)甲、乙、丙都不正确答( )6. 把由1开始自然数依次写下去，直写道第198位为止：123456789101112位那么这个数用9除的余数是(A)4 (B)6 (C)7 (D)非上述答案答( )E O D CBA 图1二、填空题1. 在三边长诗连续自然数，周长不超过100的三角形中，锐角三角形的个数是__________.2. 设a ，b ，c 分别是A B C 的三个角A ∠，B ∠，C ∠所对边的长，而且60A ∠= ，那么cba b a c +++的值是__________.3. []a 表示不大于数a 的最大整数.例如21⎡⎤=⎣⎦，22⎡⎤-=-⎣⎦，那么方程[]13122x x +=-的所有根的和是__________.4. 设自然数n 有下面性质：从1，2，…，n 中任取50个不同的数，这50个数中必有两个数之差等于7，这样的n 最大的一个是__________.5. 有五位数的正奇数x ，将x 中的所有2都换成5，所有5都换成2，其他数字不变，得到一个新的五位数，记作y .若x 和y 满足等式21y x =+，那么x 是__________.第二试一、当a 、b 为何值时，方程()()2222134420x a x a ab b ++++++=有实根？二、已知：D 是A B C 的边A C 上的一点，且:2:1A C C D =，45C ∠= ，60ADB ∠= ，求证：A B 是B C D 的外接圆的切线.三、已知存在正整数n ，能使数1111n个被1987整除，求证：数 1111n p = 个 9999n 个 8888n 个 7777n 个和11111n q += 个 19999n + 个 18888n +个都能被1987整除.。

1987年全国初中数学联合竞赛试题及解答

A．可以是直线 AD 上的某一点 C．只是线段 AD 的中点
3．如图，AB 是半圆的直径，O 是圆心，CD⊥AB，DE⊥OC，如果 AD，BD 和 CD 的长都是有理数，那么命题甲：OE 的长是有理数．乙：DE 的长是有理数．丙：图中所有字母表示的线段的长都是有理数．（ A．只有甲正确 C．只有甲、乙正确 B．只有乙正确 D．甲、乙、丙都正确））
OD ． OD 与 BC 相交于 E ．
A
D B O E C
易得 BOD 2BCD 90 ．因为 DBC ADB ACB 15 ，所以 DOC 30 ，所以 BOC 120 ．因为 OB OC ，所以 OBC OCB
180 120 30 ． 2
C E A D O B
4．已知方程 | x | ax 1 有一个负根而且没有正根，那么 a 的取值范围是（ A． a 1 B． a 1 C． a ≥ 1 D．非上述答案
5．已知四边形 ABCD 内有一点 E，连接 AE，BE，CE，DE，将四边形 ABCD 分成四个面积相等的三角形，那么命题甲：ABCD 是凸四边形．乙： E 是对角线 AC 的中点或对角线 BD 的中点．丙： ABCD 是平行四边形．在以上三条中（ A．只有甲正确） B．只有乙正确 C．甲、乙、丙都正确 D．甲、乙、丙都不正确，那么这个
∵ 4[(1 a2 ) (a 2b)2 ] ≤ 0 ，
7
∴ 4[(1 a)2 (a 2b)2 ] 0 ， ∴ 1 a 0 ，且 a 2b 0 ，即 a 1， b
1 ． 2 1 时方程有实数根． 2
因此，当 a 1 ， b
二、【解析】证法 1：如图 3，作 △B 设圆心为 O ，边 OB ，OC ， C D 的外接圆，

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232Tools for Teaching1994
1987:The Parking Lot Problem The owner of a paved,100-ft-by-200-ft,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 realize that squeezing as many cars into the lot as possible leads to right-angle parking with the cars aligned side by side.However,inexpe-rienced drivers have difﬁculty parking their cars this way,which can give rise to expensive insurance claims.To reduce the likelihood of damage to parked vehicles,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 that can be accommodated in the lot,leading to less revenue for the parking lot owner. Comments by the Contest Director
The problem was contributed by Maurice D.Weir(Mathematics Dept., Naval Postgraduate School,Monterey,CA).
Of the156papers entered in the Contest,125dealt with this problem. The Outstanding papers were by teams from Calvin College and Rensse-laer Polytechnic Institute,both of which concluded that the lot could be designed to hold more than75average-sized cars.Their papers,together with commentaries,were published as follows:
Bingle,Richard,Dan Meindertsma,and William Oostendorp.1988.De-signing the optimal placement of spaces in a parking lot.The UMAP Journal9(1)(1988):13–35.
Special Issue:Mathematical Competition in Modeling.1987.Mathematical Modeling:An International Journal9(10):765–784.This special issue contains all four Outstanding papers.。