2000-2013美国数学建模竞赛(MCM、ICM)历年试题汇总(1)

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。

Problem 1Aunt Anna is years old. Caitlin is years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?安妮今年42岁，凯琳比柏娜小五岁，而柏娜的年龄是安妮的一半。

试问凯琳今年几岁？Problem 2Which of these numbers is less than its reciprocal?下列那一个数小于它的倒数？Problem 3How many whole numbers lie in the interval between and 有多少个整数介于5/3和2π之间？Problem 4In only of the working adults in Carlin City worked at home. By the "at-home" work force increased to . In there wereapproximately working at home, and in there were . The graph that best illustrates this is在卡林市，1960年只有5%的成年工作者在家工作，至1970年在家工作人数增加到8%，1960年大约有15%的人在家工作，而在1990年则有30%。

试问下面那一个图是这种情形的最佳说明Problem 5Each principal of Lincoln High School serves exactly one -year term. What is the maximum number of principals this school could have during an -year period?林肯中学每一位校长都洽服务一次三年任期，则在8年期间林肯中学最多有几位校长？Problem 6Figure is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is右图ABCD是正方形。

近几年美国大学生数学建模竞赛（USMCM）的题目包括：
2019年：建立一个模型来模拟东海和黄海的湍流。

2018年：预测联合国安理会和联合国大会决策结果及党派之间的关系。

2017年：建立一个模型来识别投资者风险偏好并帮助他们优化投资组合。

2016年：建立一个模型来识别用户a浏览网页时的行为特征，以便更好地理解和预测用户的行为。

2015年：建立一个模型，根据通信终端的传输速率，识别用户的实时视听传输需求。

2014年：建立一个模型来模拟社会文化传播的影响。

2013年：建立一个模型，根据用户的行为来预测新闻传播的趋势，并建议相关策略。

2012年：建立一个模型来优化公共汽车系统，以满足不同地区乘客的旅行需求。

2011年：建立一个模型，根据居民就医环境的不同，构建卫生保健系统的合理结构。

2010年：建立一个模型，预测印度洋及其邻近海域的风暴强度，以及其对当地的影响。

地球资源的消耗速度快,越来越多的人关注人类社会的未来。

自1960年以来，已经有许多专家研究可持续发展。

然而大多数人的研究对象是整个世界，一个国家或一个地区。

几乎没有人选择48个最不发达国家(LDC)在联合国为研究对象列表。

然而，LDC国家集团共享许多相同的点.他们的发展道路也有法律的内涵。

本文选择这些国家为研究对象针对发现常规的可持续发展道路。

本文组织如下。

第二部分介绍研究的背景和本研究的意义。

第三节描述了我们对可持续发展的理解细节和显示我们的评估系统的建立过程和原理，那么我们估计每一个国家的LDC和获得可持续发展的能力和等级。

第四节提供了一个最糟糕的国家毛里塔尼亚计划指数在第三节。

第五节演示了在第四节的合理性和可用性计划。

最后在第六节总结本文的主要结论和讨论的力量和潜在的弱点。

地球上的资源是有限的。

三大能源石油、天然气和煤炭可再生。

如何避免人类的发展了资源枯竭和实现可持续发展目标是现在的一个热门话题。

在过去的两个世纪，发达国家已经路上,先污染,再控制和达到高水平的可持续发展。

发展中国家希望发展和丰富。

然而,因为他们的技术力量和低水平的经济基础薄弱，浪费和低效率的发展在这些国家是正常的.所以本文主要关注如何帮助发展中国家特别是48在联合国最不发达国家实现可持续发展是列表可持续发展的理解是解决问题的关键。

可持续发展的定义经历了一个长期发展的过程.在这里,布伦特兰可持续发展委员会的简短定义的”能力发展可持续- - — - - -以确保它既满足现代人的需求又不损害未来的能力代来满足自己的需求”[1]无疑是最被广泛接受的一个在各种内吗定义。

这个定义方面发挥了重要作用在很多国家的政策制定的过程。

然而,为了证明一个国家的现状是否可持续不可持续的,更具体的定义是必要的更具体的概念，我们认为,如果一个国家的发展是可持续的，它应该有一个基本的目前的发展水平，一个平衡的国家结构和一个光明的未来.基本的发展水平反映了国家的基础和潜力。

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.。

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。

中国水资源战略摘要Summary为了确定中国最佳的水资源战略，将中国分为九大流域，首先借助MATLAB建立多项式拟合模型来预测出中国2013年到2025年每年各流域的供水量和需水量，接着在可持续发展的原则指导下建立区域水资源合理配置模型，对每一个流域，采用水资源综合短缺度最小为目标函数, 对地表水、地下水等多种水源统筹考虑, 用权重区别对待工业、农业、生活、生态环境等不同领域的用水需求, ,从而求出各个流域最小的缺水量。

再根据前面的两个模型所预测出来的各流域的缺水量，建立最佳的补水模型解决缺水问题：通过对实际问题的分析，可能的补水方案有两个：方案一是直接从珠江流域调水到缺水的流域，方案二是沿海流域采取海水淡化补水，内陆流域采取直接从珠江流域调水过去，经过分析、计算发现方案二是最佳的。

最后，我们统筹考虑我们所制定的水策略，发现其无论是对经济、社会还是生态环境都将产生重大影响。

In order to determine the best water resources strategy, we divided China into nine basins. Firstly, we established polynomial fitting model with the use of MATLAB to predict the water supply and the water demand of every basin from 2013 to 2025. Secondly, we established the regional water resources rational allocation model under the guidance of the principle of sustainable development. In this model, through taking the minimum comprehensive water shortage degree as objective , surface water , groundwater and other water are considered, and different weightings are used for industrial, agricultural, domestic and ecological water users in order to realize regional water resources rational allocation .In this way can we obtained the minimum amount of water scarcity in every basin. Thirdly, according to the data predicted based on the previous two models, we can establish the optimal replenishment model to solve the problem of water shortage. We identified two possible replenishment program based on the analysis of the actual problems. One is to transfer the water of the Pearl River to basins where lack of water resources, another is to transfer the water of the Pearl River to inland basins directly while we meet the water shortage of coastal basins by desalination. After analysis and calculation, we find second program is the best. Finally, we find the water strategy we developed has a significant impact on the economic, social and ecological environment after we considered the models we established.关键字：水策略多项式拟合模型区域水资源合理配置模型补水模型Keywords：Water strategythe Polynomial fitting modelThe Regional water resources rational allocation modelthe Replenishment model§1.问题重述Problem restatement水是生命之源, 是人类生存和发展不可替代的资源, 是经济、社会可持续发展的基础。

2003 MCM ProblemsPROBLEM A: The Stunt PersonAn exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! A stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by camera, etc.).Your job is to:•determine what size boxes to use•determine how many boxes to use•determine how the boxes will be stacked•determine if any modifications to the boxes would help•generalize to different combined weights (stunt person & motorcycle) and different jump heightsNote that, in "Tomorrow Never Dies", the James Bond character on a motorcycle jumps over a helicopter.PROBLEM B: Gamma Knife Treatment PlanningStereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators.The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. All 201 beams simultaneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed a “shot.” Shots can be represented as different spheres. Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume is a bounded, three-dimensional digital image that usually consists of millions of points.The goal of radiosurgery is to deplete tumor cells while preserving normal structures.Since there are physical limitations and biological uncertainties involved in this therapy process, a treatment plan needs to account for all those limitations and uncertainties. In general, an optimal treatment plan is designed to meet the following requirements.1.Minimize the dose gradient across the target volume.2.Match specified isodose contours to the target volumes.3.Match specified dose-volume constraints of the target and critical organ.4.Minimize the integral dose to the entire volume of normal tissues or organs.5.Constrain dose to specified normal tissue points below tolerance doses.6.Minimize the maximum dose to critical volumes.In gamma unit treatment planning, we have the following constraints:1.Prohibit shots from protruding outside the target.2.Prohibit shots from overlapping (to avoid hot spots).3.Cover the target volume with effective dosage as much as possible. But at least90% of the target volume must be covered by shots.e as few shots as possible.Your tasks are to formulate the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution. While designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient.2002 Contest ProblemsProblem AAuthors: Tjalling YpmaTitle: 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.Problem BAuthors: Bill Fox and Rich WestTitle: 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 BHeightened security at and around airportsPassengers' fearLoss 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, and consider alternatives for handling "bumped" passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.MCM2000Problem 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 analyst at the FAA r traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA has posed the following problemsRequirement A: Given two airplanes flying in space, when should the air traffic controller ld the air traffic controller consider the objects to be too close and to require intervention?Requirement B: An airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of 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 sector (1) at any one instant? (2) During any given interval of time? (3) During 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 conclusionsProblem 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 grid (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 allotted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1, 2, 3, ... . Each transmitter will 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 assign 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 interferenceRequirement A: There are several constraints on frequency assignments. First, no two transmitters within distance 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 constraints, what can we say about the span in,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 differ by at least some given integer k, while those at distance at most must still differ by at least one. What can 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 findingsMCM2000问题A 空间交通管制为加强安全并减少空中交通指挥员的工作量，联邦航空局(FAA)考虑对空中交通管制系统添加软件，以便自动探测飞行器飞行路线可能的冲突，并提醒指挥员。

美赛历年题目汇总
以下是美赛历年的一些题目汇总：
2018年的题目是“多跳HF无线电传播语言传播趋势”；
2017年的题目是“管理赞比西河高速路收费合并”；
2016年的题目是“浴缸的水温模型解决空间碎片问题”；
2015年的题目是“根除病毒寻找失踪的飞机”；
2014年的题目是“(交通流、路况)优化(体育教练)综合评价”；
2013年的题目是“平底锅受热，热力学、几何(大模型解答所有题目)，可利用淡水资源的匮乏，(水资源)预测、最优化”；
2012年的题目是“一棵树的叶子沿着BigLongRiver野营，(流程)优化”；
需要注意的是，这里只列出了部分美赛历年的题目，而且每年的题目都可能有所不同。

同时，美赛赛题的难度较高，需要具备一定的数学建模和计算机编程能力。

因此，在参加美赛前，建议充分准备，提高自己的数学建模和计算机编程能力。

PROBLEM A: The Ultimate Brownie Pan问题A：终极布朗尼蛋糕盘When 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).当蛋糕在矩形盘里加热时，热量会集中在蛋糕的四个角导致加热过度（在较小的边缘处也有这种情况）。

In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges.在圆形的盘子里，热量均匀的分布在蛋糕表面，所以不会加热过度。

However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven.然而，由于大多数的烤箱是矩形的，使用的盘子是圆形的，不能有效的利用烤箱里的空间。

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.建立模型来显示在使用不同形状的盘子时（方形，圆形和两者之间的其他形状），热量在其表面的分布情况。

Assume假设1. A width to length ratio of W/L for the oven which is rectangular in shape.1，烤箱是矩形的，长为l，宽为W。

2000 AMC 10 ProblemsProblem 1In the year 2001, the United States will host the International Mathematical Olympiad. Let , , and be distinct positive integers such that the product 2001=••O M I . What is the largest possible value of the sum O M I ++ ?（A ）23 （B ）55 （C ）99 （D ）111 （E ）671Problem 22000 （20002000）＝（A ）20012000 （B ）20004000 （C ）40002000 （D ）2000000,000,4 （E ）000,000,42000Problem 3Each day, Jenny ate 20% of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, 32 remained. How many jellybeans were in the jar originally?（A ）40 （B ）50 （C ）55 （D ）60 （E ）75Problem 4Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was , but in January her bill was because she used twice as much connect time as in December. What is the fixed monthly fee?（A ）2.53 （B ）5.06 （C ）6.24 （D ）7.42 （E ）8.77Problem 5Points M and N are the midpoints of sides PA and PB of △PAB. As P moves along a line that is parallel to side AB, how many of the four quantities listed below change? (a) the length of the segment MN (b) the perimeter of △PAB(c) the area of △PAB (d) the area of trapezoid ABNM（A ）0 （B ）1 （C ）2 （D ）3 （E ）4Problem 6The Fibonacci sequence 1，1，2，3，5，8，13，21，…… starts with two s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?（A ）0 （B ）4 （C ）6 （D ）7 （E ）9Problem 7 In rectangle , , is on , and and trisect .What is the perimeter of ?（A ）333+ （B ）3342+ （C ）222+ （D ）2533+ （E ）3352+ Problem 8At Olympic High School, 52 of the freshmen and 54 of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?（A ）There are five times as many sophomores as freshmen.（B ）There are twice as many sophomores as freshmen.（C ）There are as many freshmen as sophomores.（D ）There are twice as many freshmen as sophomores.（E ）There are five times as many freshmen as sophomores.Problem 9If , where , then（A ）－2 （B ）2 （C ）2－2p （D ）2p-2 （E ）｜2p-2｜Problem 10The sides of a triangle with positive area have lengths , , and . The sides of a second triangle with positive area have lengths , , and . What is the smallest positive number that is not a possible value of ｜x-y ｜?（A ）2 （B ）4 （C ）6 （D ）8 （E ）10Problem 11Two different prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?（A ）21 （B ）60 （C ）119 （D ）180 （E ）231Problem 12Figures 0, 1, 2 and 3 consist of 1, 5, 13, and 25 nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be infigure 100?Figure 0 Figure 1 Figure 2 Figure 3（A ）10401 （B ）19801 （C ）20201 （D ）39801 （E ）40801 Problem 13There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?Problem 14Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71, 76, 80, 82, and 91. What was the last score Mrs. Walter entered?（A ）71 （B ）76 （C ）80 （D ）82 （E ）91Problem 15Two non-zero real numbers, and , satisfy. Find a possible value of ab ab b a -+ . （A ）-2 （B ）-21 （C ）31 （D ）21 （E ）2 Problem 16The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment AB meets segment CD at E. Find the length of segment AE.（A ）354 （B ）355 （C ）7512 （D ）52 （E ）9565Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?（A ）<dollar/>3.63 （B ）<dollar/>5.13 （C ）<dollar/>6.30 （D ）<dollar/>7.45 （E ）<dollar/>9.07Problem 18Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?（A ）24 （B ）27 （C ）39 （D ）40 （E ）42Problem 19Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is times the area of the square. The ratio of the area of the other small right triangle to the area of the square is（A ）121 m （B ）m （C ）1-m （D ）m 41 （E ）281mProblem 20Let A, M, and C be nonnegative integers such that A+M+C=10. What is the maximum value of A ·M ·C + A ·M + M ·C +C ·A?（A ）49 （B ）59 （C ）69 （D ）79 （E ）89Problem 21If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?I. All alligators are creepy crawlers.II. Some ferocious creatures are creepy crawlers.III. Some alligators are not creepy crawlers.（A ）I only （B ）II only （C ）III only （D ）II and III only （E ）None must be true Problem 22One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?（A ）3 （B ）4 （C ）5 （D ）6 （E ）7When the mean, median, and mode of the list 10, 2, 5, 2, 4, 2, x are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ? （A ）3 （B ）6 （C ）9 （D ）17 （E ）20Problem 24Let be a function for which. Find the sum of all values of for which. （A ）-31 （B ）-91 （C ）0 （D ）95 （E ）35 Problem 25In year , the day of the year is a Tuesday. In year, the day is also a Tuesday. On what day of the week did the day of year occur?2000 AMC 10 SolutionProblem 1 The following problem is from both the 2000 AMC 12 #1 and 2000 AMC 10 #1, so both problems redirect to this page.The sum is the highest if two factors are the lowest.So, 1·3·776=2001 and 1+3+667=671 (E)Problem 2 The following problem is from both the 2000 AMC 12 #2 and 2000 AMC 10 #2, so both problems redirect to this page.2000 （20002000）＝12000（20002000）=20012000 (A)Problem 3 The following problem is from both the 2000 AMC 12 #3 and 2000 AMC 10 #3, so both problems redirect to this pageSince Jenny eats 20% of her jelly beans per day, 80%=4/5 of her jelly beans remain after one day. Let be the number of jelly beans in the jar originally.325454=••x x=50 (B) Problem 4Let be the fixed fee, and be the amount she pays for the minutes she used in the first month.x+y=12.48 x+2y=17.54 y=5.06 x=7.42 We want the fixed fee, which is (D) Problem 5(a) Clearly AB does not change, and MN=0.5AB, so MN doesn't change either. (b) Obviously, the perimeter changes.(c) The area clearly doesn't change, as both the base AB and its corresponding height remain the same.(d) The bases AB and MN do not change, and neither does the height, so the area of the trapezoid remains the same.Only quantity changes, so the correct answer is .Problem 6 The following problem is from both the 2000 AMC 12 #4 and 2000 AMC 10 #6, so both problems redirect to this page.Note that any digits other than the units digit will not affect the answer. So to make computation quicker, we can just look at the Fibonacci sequence in :The last digit to appear in the units position of a number in the Fibonacci sequence is 6 (C).Problem 7AD=1 Since ∠ADC is trisected, ∠ADP=∠PDB=∠BDC=30º Thus, PD=332 BD=2 BP=332333=- Adding, 3342+Problem 8Let be the number of freshman and be the number of sophomores.s f 5452= f=2s There are twice as many freshmen as sophomores. Problem 9 The following problem is from both the 2000 AMC 12 #5 and 2000 AMC 10 #9, so both problems redirect to this page.When x<2, x-2 is negative so ∣x -2∣=2-x =p and x=2-pThus x-p=(2-p)-p=2-2p (C)Problem10From the triangle inequality, 2<x <10 and 2<y <10. The smallest positive number not possible is 10-2, which is . (D)Problem 11 The following problem is from both the 2000 AMC 12 #6 and 2000 AMC 10 #11, so both problems redirect to this page.All prime numbers between 4 and 18 have an odd product and an even sum. Any odd number minus an even number is an odd number, so we can eliminate B andD. Since the highest two prime numbers we can pick are 13 and 17, the highest number we can make is (13) (17) – (13+17) = 221-30=191. Thus, we can eliminateE. Similarly, the two lowest prime numbers we can pick are 5 and 7, so the lowest number we can make is (5) (7) – (5+7) =23. Therefore, A cannot be an answer. So, the answer must be .Problem 12 The following problem is from both the 2000 AMC 12 #8 and 2000 AMC 10 #12, so both problems redirect to this page.Solution 1We have a recursion: .I.E. we add increasing multiples of each time we go up a figure. So, to go from Figure 0 to 100, we addWe then add to the number of squares in Figure 0 to get, which ischoice Solution 2We can divide up figure to get the sum of the sum of the first odd numbers and the sum of the first odd numbers. If you do not see this, here is the example for :The sum of the first odd numbers is 2n , so for figure , thereare 22)1(n n ++ unit squares. We plug in n=100 to get 20201, which is choice Solution 3Using the recursion from solution 1, we see that the first differences of 4,8,12, … form an arithmetic progression, and consequently that the second differences are constant and all equal to . Thus, the original sequence can be generated from a quadratic function.If c bn an n f ++=2)(, and f(0)=1, f(1)=5, and f(2)=13, we get a system of three equations in three variables:f(0)=1 gives c=1; f(1)=5 gives a+b+c=5; f(2)=13 gives 4a+2b+c=13 Plugging in into the last two equations gives a+b=4 4a+2b=12 Dividing the second equation by 2 gives the system: a+b=4 2a+b=6Subtracting the first equation from the second gives , and hence . Thus, our quadratic function is: 122)(2++=n n n fCalculating the answer to our problem, f(100)=20201Problem 13In each column there must be one yellow peg. In particular, in the rightmost column, there is only one peg spot, therefore a yellow peg must go there.In the second column from the right, there are two spaces for pegs. One of them is in the same row as the corner peg, so there is only one remaining choice left forthe yellow peg in this column.By similar logic, we can fill in the yellow pegs as shown:After this we can proceed to fill in the whole pegboard, so there is only arrangement of the pegs. The answer is (B)Problem 14 The following problem is from both the 2000 AMC 12 #9 and 2000 AMC 10 #14, so both problems redirect to this page.Solution 1The first number is divisible by 1.The sum of the first two numbers is even.The sum of the first three numbers is divisible by 3.The sum of the first four numbers is divisible by 4.The sum of the first five numbers is 400.Since 400 is divisible by 4, the last score must also be divisible by 4. Therefore, the last score is either 76 or 80.Case 1: 76 is the last number entered.Since 400≡76≡1 (mod 3), the fourth number must be divisible by 3, but none of the scores are divisible by 3. Case 2: 80 is the last number entered. Since 80≡2 (mod 3), the fourth number must be 2 (mod 3). Thatnumber is 71 and only 71. The next number must be 91, since the sum of the first two numbers is even.So the only arrangement of the scores 76, 82, 91,71,80Solution 2We know the first sum of the first three numbers must be divisible by 3, so we write out all 5 numbers (mod 3), which gives 2,1,2,1,1, respectively. Clearly the only way to get a number divisible by 3 by adding three of these is by adding the three ones. So those must go first. Now we have an odd sum, and since the next average must be divisible by 4, 71 must be next. That leaves 80 for last, so the answer is .Problem 15 The following problem is from both the 2000 AMC 12 #11 and 2000 AMC 10 #15, so both problems redirect to this page.22)()()(22222=-=----+=--+=-+ba ab b a b a b a b a b a ab b a ab a b b a (E)Alternatively, we could test simple values, like )21,1(),(=b a , which would yield 2=-+ab ab b a Another way is to solve the equation for giving 1+=a a b , then substituting this into the expression and simplifying gives the answer ofProblem 16Solution 1Let be the line containing A and B and let be the line containing C and D. If we set the bottom left point at (0,0), then A=(0,3), B=(6,0), C=(4,2), and D=(2,0) . The line is given by the equation 11b x m y +=. The -intercept is A=(0,3), so 1b =3. We are given two points on , hence we can compute the slope, to be 210630-=-- , so is the line 321+-=x y Similarly, is given by 22b x m y +=. The slope in this case is 12402=--, so 2b x y +=. Plugging in the point (2,0) gives us 2b =-2, so is the line 2-=x y . At E, the intersection point, both of the equations must be true,2-=x y , 321+-=x y so 3212+-=-x x SO 310=x 34=y We have the coordinates of and , so we can use the distance formula here: 355)334()0310(22=-+- which is answer choiceSolution 2Draw the perpendiculars from andto , respectively. As it turns out, . Let be the point onfor which . , and, so by AA similarity,By the Pythagorean Theorem, wehave, ,。

B 题 钢管订购和运输
要铺设一条1521A A A →→→ 的输送天然气的主管道, 如图一所示(见下页)。

经筛选后可以生产这种主管道钢管的钢厂有721,,S S S 。

图中粗线表示铁路，单细线表示公路，双细线表示要铺设的管道(假设沿管道或者原来有公路，或者建有施工公路)，圆圈表示火车站，每段铁路、公路和管道旁的阿拉伯数字表示里程(单位km)。

为方便计，1km 主管道钢管称为1单位钢管。

一个钢厂如果承担制造这种钢管，至少需要生产500个单位。

钢厂i S 在指定期限内能生产该钢管的最大数量为i s 个单位，钢管出厂销价1单位钢管为i p 万元，如下表：
1单位钢管的铁路运价如下表：
1000km 以上每增加1至100km 运价增加5
公路运输费用为1单位钢管每公里0.1万元（不足整公里部分按整公里计算）。

钢管可由铁路、公路运往铺设地点（不只是运到点1521,,,A A A ，而是管道全线）。

（1）请制定一个主管道钢管的订购和运输计划，使总费用最小（给出总费用)。

（2）请就（1）的模型分析：哪个钢厂钢管的销价的变化对购运计划和总费用影响最大，哪个钢厂钢管的产量的上限的变化对购运计划和总费用的影响最大，并给出相应的数字结果。

（3）如果要铺设的管道不是一条线，而是一个树形图，铁路、公路和管道构成网络，请就这种更一般的情形给出一种解决办法，并对图二按（1）的要求给出模型和结果。

7
7。

2013年美国数学竞赛AMC12A真题Square has side length . Point is on , and the area of is . What is ?Problem 2A softball team played ten games, scoring , and runs. They lost by one run inexactly five games. In each of the other games, they scored twice as many runs as their opponent.How many total runs did their opponents score?Problem 3A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?Problem 4What is the value ofTom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their tripTom paid $, Dorothy paid $, and Sammy paid $. In order to share the costs equally, Tomgave Sammy dollars, and Dorothy gave Sammy dollars. What is ?Problem 6In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She wassuccessful on of her three-point shots and of her two-point shots. Shenilleattempted shots. How many points did she score?Problem 7The sequence has the property that every term beginning with the third is the sumof the previous two. That is,Suppose that and . Whatis ?Problem 8Given that and are distinct nonzero real numbers such that , what is ?Problem 9In , and . Points and are on sides , , and ,respectively, such that and are parallel to and , respectively. What is the perimeter of parallelogram ?Problem 10Let be the set of positive integers for which has the repeating decimalrepresentation with and different digits. What is the sum of the elements of ?Problem 11Triangle is equilateral with . Points and are on and points and areon such that both and are parallel to . Furthermore, triangle andtrapezoids and all have the same perimeter. What is ?Problem 12The angles in a particular triangle are in arithmetic progression, and the side lengths are . Thesum of the possible values of x equals where , and are positive integers. Whatis ?Let points and . Quadrilateral is cut into equalarea pieces by a line passing through . This line intersects at point , where these fractions are in lowest terms. What is ?Problem 14The sequence, , , ,is an arithmetic progression. What is ?Problem 15Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?Problem 16, , are three piles of rocks. The mean weight of the rocks in is pounds, the mean weight ofthe rocks in is pounds, the mean weight of the rocks in the combined piles and is pounds, and the mean weight of the rocks in the combined piles and is pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles and ?Problem 17A group of pirates agree to divide a treasure chest of gold coins among themselves as follows.The pirate to take a share takes of the coins that remain in the chest. The number of coinsinitially in the chest is the smallest number for which this arrangement will allow each pirate to receivea positive whole number of coins. How many coins does the pirate receive?Problem 18Six spheres of radius are positioned so that their centers are at the vertices of a regular hexagon of side length . The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent tothe larger sphere. What is the radius of this eighth sphere?Problem 19In , , and . A circle with center and radius intersects atpoints and . Moreover and have integer lengths. What is ?Problem 20Let be the set . For , define to mean thateither or . How many ordered triples of elements of have theproperty that , , and ?Problem 21Consider . Which of the following intervals contains ?A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome is chosen uniformly at random. What is the probability that is also a palindrome?Problem 23is a square of side length . Point is on such that . The square regionbounded by is rotated counterclockwise with center , sweeping out a region whose areais , where , , and are positive integers and . What is ?Problem 24Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?Problem 25Let be defined by . How many complex numbers are there suchthat and both the real and the imaginary parts of are integers with absolute value atmost ?Answer Key1. E2. C3. E4. C5. B6. B7. C8. D9. C10.D11.C12.A13.B14.B15.D16.E17.D18.B19.D20.B21.A22.E23.C24.E25.A。

amc8历年真题答案解析【Amc8历年真题答案解析】Amc8，全名为American Mathematics Competition 8，是一项面向美国中学生的数学竞赛。

每年11月份举行的Amc8竞赛，旨在通过一系列挑战性的数学问题，培养学生的数学思维能力、创造力和解决问题的能力。

以下是Amc8历年真题的答案解析。

一、2000年真题：1. 答案：C解析：问题要求选择一个在100和999之间的正整数，满足各个数字的平方和等于该数本身。

首先明确数字的范围是100到999，因此只需要考虑三位数。

假设一个三位数为$abc$，那么它满足条件$a^2+b^2+c^2=abc$。

考虑到$a,b,c$的范围在1到9之间，可以通过穷举法逐个尝试。

当$a=2$时，$b=5$和$c=1$，满足条件$2^2+5^2+1^2=251$，因此答案为C。

2. 答案：D解析：问题给出了一个小测试，Tom回答7道题目，其中4道题回答正确。

考虑到每道问题只有两个选择（正确或错误），因此回答7道题目的所有可能性为$2^7=128$。

在这些可能性中，只有一种情况Tom回答了4道题目正确，因此答案为D。

3. 答案：A解析：问题给出了一个图形，包含了一系列长方形。

我们需要计算图形中所有（不重叠）长方形的个数。

首先考虑长方形最小边长为1的情况，可以在图形中找到4个这样的长方形。

然后考虑最小边长为2的情况，可以找到3个长方形。

以此类推，最小边长为3、4、5的情况下，分别可以找到2个、1个、0个长方形。

因此，总共可以找到的长方形个数为$4+3+2+1=10$，答案为A。

4. 答案：C解析：问题给出了一个5乘5的正方形网格，要求从左上角走到右下角，只能向右或向下移动，且不能重复经过同一格。

我们需要计算从左上角到右下角有几条路径。

可以发现，从左上角到右下角，一共需要移动4次向右，4次向下。

因此可以将问题转化为在8个移动中选择4个向右，即$C_8^4=\frac{8!}{4!4!}=70$，答案为C。

数学建模大赛历年试题1．MCM（美国大学生数学建模竞赛）1985 A题动物群体管理1985 B题战略物资存储管理1986 A题水道测量数据1986 B题应急设施的位置1987 A题盐的贮存1987 B题停车场1988 A题确定走私船的位置1988 B题两辆铁路平板车的装货问题1989 A题蠓的分类1989 B题飞机排队1990 A题药物在大脑中的分布1990 B题扫雪问题1991 A题估计水箱的流水量1991 B题最小费用极小生成树1992 A题航空控制雷达的功率1992 B题应急电力修复系统1993 A题加速餐厅剩菜堆肥的生成1993 B题倒煤台的操作方案1994 A题建筑费用1994 B题计算机传输1995 A题单螺旋线1995 B题教师薪金分配1996 A题海底探测1996 B题竞赛论文的评定1997 A题疾走龙属问题1997 B题开会决策1998 A题MRI扫描仪1998 B题学生等级划分1999 A题小型星撞击1999 B题非法集会1999 C题大地污染2000 A题空中交通控制2000 C题大象的数量2002 A题风和喷水池2002 B题航空公司超员订票2003 A题特技人员2003 B题GAMMA刀治疗计划2004 A题指纹是独一无二的吗？2004 B题更快的快通系统2．CUMCM（全国大学生数学建模竞赛）1993年A题非线性交调的频率设计1993年B题球队排名问题1994年A题逢山开路1994年B题锁具装箱1995年A题一个飞行管理模型1995年B题天车与冶炼炉的作业调度1996年A题最优捕鱼策略1996年B题节水洗衣机1997年A题零件的参数设计1997年B题截断切割1998年A题投资的收益和风险1998年B题灾情巡视路线1999年A题自动化车床管理1999年B题钻井布局2000年A题DNA序列分类2000年B题钢管定购和运输2001年A题血管的三维重建2001年B题公交车调度2002年A题车灯线光源的优化设计2002年B题彩票中的数学2003年A题SARS的传播2003年B题露天矿生产的车辆安排2004年A题奥运会临时超市网点设计2004年B题电力市场的输电阻塞管理。

2000到2012年A M C10美国数学竞赛全美中学数学分级能力测验(AMC 10)2000年 第01届 美国AMC10 (2000年2月 日 时间75分钟)1. 国际数学奥林匹亚将于2001年在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ?M ?O =2001，则试问I ?M ?O 之最大值为 。

(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为 。

(A) 20002001 (B) 40002000 (C) 20004000 (D) 40000002000 (E) 200040000003. Jenny 每天早上都会吃掉她所剩下的聪明豆的20%，今知在第二天结束时，有32颗剩下，试问一开始聪明豆有 颗。

754. Candra 她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是 元。

(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直 在线移动时，下列各数值有 项会变动。

(a) MN 长 (b) △PAB 之周长 (c) △PAB 之面积 (d) ABNM 之面积(A) 0项 (B) 1项 (C) 2项 (D) 3项 (E) 4项6. 费氏数列是以两个1开始，接下来各项均为前两项之和，试问在P 0 A 0B 0C 0D 图3图2 图1 图0 费氏数列各项的个位数字中， 最后出现的阿拉伯数字为 。

(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分 ?ADC ，试问△BDP 之周长为 。

(A) 3?33(B) 2?334 (C) 2?2 (D) 2533+ (E) 2?3358. 在奥林匹克高中，有52的新生与54的高二生参加AMC 10年级测验。

美赛题目参考答案美赛题目参考答案在数学建模竞赛中，美国大学生数学建模竞赛（MCM/ICM）是其中最具代表性和影响力的比赛之一。

每年，数以千计的学生参加这一竞赛，争夺着优胜奖项。

本文将探讨美赛题目的参考答案，并分析其中的解题思路和方法。

首先，我们来看一道典型的美赛题目：假设你是一家电商公司的数据分析师，负责分析用户的购物习惯。

你需要设计一个算法，根据用户的购买历史，预测他们未来的购买行为。

请给出你的解决方案。

针对这个题目，我们可以从以下几个方面进行分析和解答。

第一，我们需要收集用户的购买历史数据。

这些数据可以包括用户的购买时间、购买金额、购买商品的类别等信息。

通过对这些数据的分析，我们可以了解用户的购买习惯和偏好。

第二，我们可以利用机器学习算法来预测用户的未来购买行为。

常用的机器学习算法包括决策树、支持向量机、随机森林等。

我们可以将用户的购买历史作为训练集，利用这些算法进行模型训练，然后利用训练好的模型对未来的购买行为进行预测。

第三，我们可以利用时间序列分析的方法来预测用户的未来购买行为。

时间序列分析是一种专门用于处理时间相关数据的方法。

通过对用户购买历史数据的时间序列进行分析，我们可以发现其中的规律和趋势，并用这些规律和趋势来预测未来的购买行为。

第四，我们可以利用关联规则挖掘的方法来预测用户的未来购买行为。

关联规则挖掘是一种用于发现数据中的频繁项集和关联规则的方法。

通过对用户购买历史数据的分析，我们可以找到一些频繁购买的商品组合，然后根据这些组合来预测用户的未来购买行为。

综上所述，针对这个题目，我们可以采用数据分析、机器学习、时间序列分析和关联规则挖掘等方法来预测用户的未来购买行为。

当然，具体的解决方案还需要根据实际情况进行调整和优化。

除了以上的解题思路，还有很多其他的方法可以用于解决这个问题。

例如，我们可以利用深度学习算法来进行预测，或者利用网络分析的方法来分析用户之间的关系等等。

不同的方法有不同的优缺点，我们可以根据具体情况选择合适的方法。