2011年美国大学生数学建模竞赛题目
MCM美国大学生数学建模比赛2000-2011年题目
2000 Mathemat ical Contest in ModelingThe ProblemsProblem A: Air traffic ControlProblem B: Radio Channel AssignmentsProblem A Air traffic ControlDedicated to the memory of Dr. Robert Machol, former chief scientist of the Federal Aviation AgencyTo 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 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: An 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 sector (1) 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.Problem BRadio 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.Figure 1An 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 interference.Requirement A: There are several constraints on 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 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 2s differ by at least some given integer k, while those at distance at most 4s 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 findings.2001Problem 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 dis k (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 theright.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 s poked 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 windspeeds for different road grades starting from zero percent to ten percent in onepercent increments. (Road grade is defined to be the ratio of the total rise of a hilldivided by the length 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 proportional to the road grade. A riderwill 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.Problem 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 a nd 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 i nland 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-bound lanes of I-26into two lanes of Columbia-bound traffic, essentially turning the entire I-26 intoone-way traffic, significantly improve evacuation traffic flow?2.In 1999, the simultaneous evacuation of the state's entire coastal region was ordered.Would the evacuation traffic flow improve under an alternative strategy that staggers the evacuation, perhaps county-by-county over some time period consistent with thepattern of how hurricanes affect the coast?3.Several smaller highways besides I-26 extend inland from the coast. Under whatconditions would it improve evacuation flow to turn around traffic on these?4.What effect would it have on evacuation flow to establish more temporary shelters inColumbia, to reduce the traffic leaving Columbia?5.In 1999, many families leaving the coast brought along their boats, campers, andmotor homes. Many drove all of their cars. Under what conditions should there berestrictions on vehicle types or numbers of vehicles brought in order to guaranteetimely evacuation?6.It has been suggested that in 1999 some of the coastal residents of Georgia and Florida,who were fleeing the earlier predicted landfalls of Hurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation traffic flow? Clearly identify what measures of performance are used tocompare strategies. Required: Prepare a short newspaper article, not to exceed twopages, 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.2002 Mathemat ical Contest in ModelingThe ProblemsProblem AAuthors: Tjalling YpmaTit le: 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 WestTit le: 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 situa tion: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.2003 MCM ProblemsPROBLEM A: The Stunt PersonAn exciting action scene in a m ovie is going to be filmed, and you are the stunt coordinator! A stunt person on a m otorcycle 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 cam era, 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 Jam es Bond character on a m otorcycle jumps over a helicopter.PROBLEM B: G amma Knife Treat ment PlanningStereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, sm all intracranial 3D brain tum or 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 beam s, and external high-energy photon beams from linear accelerators.The gamma knife unit delivers a single high dose of ionizing radiation emanating from201 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 diff erent 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 volum e larger than one shot, m ultiple shots can be used to cover the entire t arget. In practice, m ost target volum es are treated with 1 to 15 shots. The target volum e is a bounded, three-dimensional digital image that usually consists of m illions of points.The goal of radiosurgery is to deplete tum or cells while preserving norma l structures. Since there are physical limitations and biological uncertainties involved in this therapy process, a treatm ent plan needs to account for all those limitations and uncertainties. In general, an optimal treat m ent plan is designed to m eet 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 treatm ent 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 least 90% ofthe target volume must be covered by shots.e as few shots as possible.Your tasks are to formulate the optim al treat m ent 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.2003 ICM ProblemPROBLEM C:To view and print problem C, you will need to have the Adobe Acrobat Reader installed in your Web browser. Downloading and installing acrobat is simple, safe, and only takes a few minutes. Download Acrobat Here.2004 MCM ProblemsPROBLEM A: Are Fingerprints Unique?It is a commonplace belief that the thumbprint of every human who has ever lived is different. Develop and analyze a model that will allow you to assess the probability that this is true. Compare the odds (that you found in this problem) of misidentification by fingerprint evidence against the odds of misidentification by DNA evidence.PROBLEM B: A Faster QuickPass System"QuickPass" systems are increasingly appearing to reduce people's time waiting in line, whether it is at tollbooths, amusement parks, or elsewhere. Consider the design of a QuickPass system for an amusement park. The amusement park has experimented by offering QuickPasses for several popular rides as a test. The idea is that for certain popular rides you can go to a kiosk near that ride and insert your daily park entrance ticket, and out will come a slip that states that you can return to that ride at a specific time later. For example, you insert your daily park entrance ticket at 1:15 pm, and the QuickPass states that you can come back between 3:30 and 4:30 pm when you can use your slip to enter a second, and presumably much shorter, line that will get you to the ride faster. To prevent people from obtaining QuickPasses for several rides at once, the QuickPass machines allow you to have only one active QuickPass at a time.You have been hired as one of several competing consultants to improve the operation of QuickPass. Customers have been complaining about some anomalies in the test system. For example, customers observed that in one instance QuickPasses were being offered for a return time as long as 4 hours later. A short time later on the same ride, the QuickPasses were given for times only an hour or so later. In some instances, the lines for people with Quickpasses are nearly as long and slow as the regular lines.The problem then is to propose and test schemes for issuing QuickPasses in order to increase people's enjoyment of the amusement park. Part of the problem is to determine what criteria to use in evaluating alternative schemes. Include in your report a non-technical summary for amusement park executives who must choose between alternatives from competing consultants.2005 MCM ProblemsPROBLEM A: Flood PlanningLake Murray in central South Carolina is formed by a large earthen dam, which was completed in1930 for power production. Model the flooding downstream in the event there is a catastrophic earthquake that breaches the dam.Two particular questions:Rawls Creek is a year-round stream that flows into the Saluda River a short distance downriver from the dam. How much flooding will occur in Rawls Creek from a dam failure, and how far back will it extend?Could the flood be so massive downstream that water would reach up to the S.C. State Capitol Building, which is on a hill overlooking the Congaree River?PROBLEM B: TollboothsHeavily-traveled toll roads such as the Garden State Parkway , Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas. Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disruption caused by the toll plazas. Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza. Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza, the flow of vehicles is required to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza. Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza. When traffic is very heavy, congestion also builds at the entry to the toll plaza because of the time required for each vehicle to pay the toll.Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effective than the current practice? Note that the definition of "optimal" is up to you to determine.2006 MCM ProblemsPROBLEM A: Posit ioning and Moving Sprinkler Systems for Irrigat ionThere are a wide variety of techniques available for irrigating a field. The technologies range from advanced drip systems to periodic flooding. One of the systems that is used on smaller ranches is the use of "hand move" irrigation systems. Lightweight aluminum pipes with sprinkler heads are put in place across fields, and they are moved by hand at periodic intervals to insure that the whole field receives an adequate amount of water. This type of irrigation sys tem is cheaper and easier to maintain than other systems. It is also flexible, allowing for use on a wide variety of fields and crops. The disadvantage is that it requires a great deal of time and effort to move and set up the equipment at regular intervals.Given that this type of irrigation system is to be used, how can it be configured to minimize the amount of time required to irrigate a field that is 80 meters by 30 meters? For this task you are asked to find an algorithm to determine how to irrigate the rectangular field that minimizes the amount of time required by a rancher to maintain the irrigation system. One pipe set is used in the field. Y ou should determine the number of sprinklers and the spacing between sprinklers, and you should find a sch edule to move the pipes, including where to move them.A pipe set consists of a number of pipes that can be connected together in a straight line. Each pipe has a 10 cm inner diameter with rotating spray nozzles that have a 0.6 cm inner diameter. When pu t together the resulting pipe is 20 meters long. At the water source, the pressure is 420 Kilo- Pascal’s and has a flow rate of 150 liters per minute. No part of the field should receive more than 0.75 cm per hour of water, and each part of the field should receive at least 2 centimeters of water every 4 days. The total amount of water should be applied as uniformly as possiblePROBLEM B: Wheel Chair Access at AirportsOne of the frustrations with air travel is the need to fly through multiple airports, and each stop generally requires each traveler to change to a different airplane. This can be especially difficult for people who are not able to easily walk to a different flight's waiting area. One of the ways that an airline can make the transition easier is to provide a wheel chair and an escort to those people who ask for help. It is generally known well in advance which passengers require help, but it is not uncommon to receive notice when a passenger first registers at the airport. In rare instances an airline may not receive notice from a passenger until just prior to landing.Airlines are under constant pressure to keep their costs down. Wheel chairs wear out and are expensive and require maintenance. There is also a cost for making the escorts available. Moreover, wheel chairs and their escorts must be constantly moved around the airport so that they are available to people when their flight lands. In some large airports the time required to move across the airport is nontrivial. The wheel chairs must be stored somewhere, but space is expensive and severely limited in an airport terminal. Also, wheel chairs left in high traffic areas represent a liability risk as people try to move around them. Finally, one of the biggest costs is the cost of holding a plane if someone must wait for an escort and becomes late for their flight. The latter cost is especially troubling because it can affect the airline's average flight delay which can lead to fewer ticket sales as potential customers may choose to avoid an airline.Epsilon Airlines has decided to ask a third party to help them obtain a detailed analysis of the issues and costs of keeping and maintaining wheel chairs and escorts available for passengers. The airline needs to find a way to schedule the movement of wheel chairs throughout each day in a cost effective way. They also need to find and define the costs for budget planning in both the short and long term.Epsilon Airlines has asked your consultant group to put together a bid to help them solve their problem. Your bid should include an overview and analysis of the situation to help them decide if you fully understand their problem. They require a detailed description of an algorithm that you would like to implement which can determine where the escorts and wheel chairs should be and how they should move throughout each day. The goal is to keep the total costs as low as possible. Your bid is one of many that the airline will consider. You must make a strong case as to why your solution is the best and show that it will be able to handle a wide range of airports under a variety of circumstances.Your bid should also include examples of how the algorithm would work for a large (at least 4 concourses), a medium (at least two concourses), and a small airport (one concourse) under high and low traffic loads. You should determine all potential costs and balance their respective weights. Finally, as populations begin to include a higher percentage of older people who have more time to travel but may require more aid, your report should include projections of potential costs and needs in the future with recommendations to meet future needs.2007 MCM ProblemsPROBLEM A: G errymanderingThe United States Constitution provides that the House of Representatives shall be composed of some number (currently 435) of individuals who are elected from each state in proportion to the state’s population relative to that of the country as a whole. While this provides a way of determining how many representatives each state will have, it says nothing about how the district represented by a particular representative shall be determined geographically. This oversight has led to egregious (at least some people think so, usually not the incumbent) district shapes that look “un natural” by some standards.Hence the following question: Suppose you were given the opportunity to draw congressional districts for a state. How would you do so as a purely “baseline” exercise to create the “simplest” shapes for all the districts in a state? The rules include only that each district in the state must contain the same population. The definition of “simple” is up to you; but you need to make a convincing argument to voters in the state that your solution is fair. As an application of your method, draw geographically simple congressional districts for the state of New Y ork.PROBLEM B: The Airplane Seat ing ProblemAirlines are free to seat passengers waiting to board an aircraft in any order whatsoever. It has become customary to seat passengers with special needs first, followed by first-class passengers (who sit at the front of the plane). Then coach and business-class passengers are seated by groups of rows, beginning with the row at the back of the plane and proceeding forward.Apart from consideration of the passengers’ wait time, from the airline’s point of view, time is money, and boarding time is best minimized. The plane makes money for the airline only when it is in motion, and long boarding times limit the number of trips that a plane can make in a day.The development of larger planes, such as the Airbus A380 (800 passengers), accentuate the problem of minimizing boarding (and deboarding) time.Devise and compare procedures for boarding and deboarding planes with varying numbers of passengers: small (85–210), midsize (210–330), and large (450–800).Prepare an executive summary, not to exceed two single-spaced pages, in which you set out your conclusions to an audience of airline executives, gate agents, and flight crews.Note: The 2 page executive summary is to be included IN ADDITION to the reports required by the contest guidelines.An article appeared in the NY Times Nov 14, 2006 addressing procedures currently being followed and the importance to the airline of finding better solutions. The article can be seen at: http://travel2.nyt /2006/11/14/business/14boarding.ht ml2008 MCM ProblemsPROBLEM A: Take a Bat hConsider the effects on land from the melting of the north polar ice cap due to the predicted increase in global temperatures. Specifically, model the effects on the coast of Florida every ten years for the next 50 years due to the melting, with particular attention given to large metropolitan areas. Propose appropriate responses to deal with this. A careful discussion of the data used is an important part of the answer.PROBLEM B: Creat ing Sudoku PuzzlesDevelop an algorithm to construct Sudoku puzzles of varying difficulty. Develop metrics to define a difficulty level. The algorithm and metrics should be extensible to a varying number of difficulty levels. You should illustrate the algorithm with at least 4 difficulty levels. Your algorithm should guarantee a unique solution. Analyze the complexity of your algorithm. Your objective should be to minimize the complexity of the algorithm and meet the above requirements.2009 MCM Problems。
美国数学建模题目2001至2012翻译
2001年A题（一）Choosing a Bicycle Wheel选择自行车车轮有不同类型的车轮可以让自行车手们用在自己的自行车上。
两种基本的车轮类型是分别用金属辐条和实体圆盘组装而成（见图1）。
辐条车轮较轻，但实体车轮更符合空气动力学原理。
对于一场公路竞赛，实体车轮从来不会用作自行车的前轮但可以用作后轮。
职业自行车手们审视竞赛路线，并且请一位识文断字的人推断应该使用哪种车轮。
选择决定是根据沿途山丘的数量和陡度，天气，风速，竞赛本身以及其他考虑作出的。
你所喜爱的参赛队的教练希望准备妥当一个较好的系统，并且对于给定的竞赛路线已经向你的参赛队索取有助于确定宜用哪种车轮的信息。
这位教练需要明确的信息来帮助作出决定，而且已经要求你的参赛队完成下面列出的各项任务。
对于每项任务都假定，同样的辐条车轮将总是装在前面，而装在后面的车轮是可以选择的。
任务1. 提供一个给出风速的表格，在这种速度下实体后轮所需要的体能少于辐条后轮。
这个表格应当包括相应于从百分之零到百分之十增量为百分之一的不同公路陡度的风速。
（公路陡度定义为一座山丘的总升高除以公路长度。
如果把山丘看作一个三角形，它的陡度是指山脚处倾角的正弦。
）一位骑手以初始速度45kph从山脚出发，他的减速度与公路陡度成正比。
对于百分之五的陡度，骑上100米车速要下降8kph左右。
任务2. 提供一个例证，说明这个表格怎样用于一条时间试验路线。
任务3. 请判明这个表格是不是一件决定车轮配置的适当工具，并且关于如何作出这个决定提出其他建议。
MCM2001B题Escaping a Hurricane's Wrath逃避飓风怒吼(一场恶风…)1999年，在Floyd飓风预报登陆之前，撤离南卡罗来纳州沿海地区的行动导致一场永垂青史的交通拥塞。
车水马龙停滞在州际公路I-26上，那是内陆上从Charleston通往该州中心Columbia相对安全处所的主要干线。
正常时轻松的两个小时驱车路要用上18个小时才能开到头。
2011数学建模(美赛)b题
Minimizing the Number of repeatersIntroductionVery high frequency (VHF) is the radio spectrum，whose frequency band ranges from 30MHz to 300MHz. VHF is always used for radio stations and television broadcasts. In addition, it is also used by signal transmission of sea navigation and aviation. Because the radio spectrum of VHF is transmitted through straight lines, a signal is easily influenced by geographical factors easily. Thus, signals become weak when it is transmitted and some low-power users need repeaters to amplify them and increase the transmission distance. We consider the situation in which every two repeaters are too close or the separate frequency is not far enough apart which can interference with each other. In order to mitigate the interference caused by the nearby repeaters, this paper employs a continuous tone-coded squelch system (CTCSS). We associate to each repeater a separate subaudible tone，that is, the subaudible tone (67Hz-250.3Hz) is added to VHF. In this way, repeaters recognize signals attached to the same subaudible tones just like secret keys. In this system, the nearby repeaters can share the same frequency pair. When users send the signals at one frequency, different repeaters with subaudible tones can recognize signals from the users the same subaudible tone. If the users in a certain area contact with each other, we should consider the signal’ s coverage area of the users and the repeaters. As long as the users’ signals are accepted by repeaters, the signals could be amplified to transmit farther. At the same time, the repeaters attached with the subaudible tones could only recognize the users with the same subaudible tones. Hence, we can consider repeaters corresponding to the number of the users, which leads to the problem of frequency channel. When the number of users in this area increases, we can add repeaters. If two repeaters have different subaudible tones, they would not communicate with each other. Thus, we should consider the problem of how the repeaters communicate with each other when they have different subaudible tones. In the mobile communication system，the spectrum is influenced by many factors such as reflex，diffraction and dispersion. Therefore, when the radio spectrum transmits in the mountainous area，we should still consider the factors above.Repeaters[4]Repeaters are a type of equipment which can amplify signals，make up the deamplification signals and support far distance communication.CTCSS[5]CTCSS(Continuous Tone Controlled Squelch System ) is short for subaudible tones, whose frequency ranges from 67Hz to 250.3Hz. It is added to the radio spectrum to make the signal carry with a unique secret key.AssumptionThe users in the area is uniform distributedThe signal of the radio spectrum in the area can’t be effected by environmentIn a certain period of time there are a small number of users removingAll repeaters have the same standardAnalysis and solution of the model to the first problemThe problem is to find a least number of repeaters in an area of radius 40 miles so that the users in this area can communicate with each other. Considering that the given area is flat, we assume that the signal ofeach repeater covers a circular area and the repeater lies in the center of the circle. The following Figure 1 shows the relationship of three adjacent repeaters.CFor case B of Figure 1, if three circles are tangent to each other, then we find that the center area cannot be covered by the singles. In order to make the signal cover the triangle area, we have to consider adding a For case C, if the intersection of three circles is not null, similar to case B, we also have to add another repeater. Thus, it is easy to find that case A, comparing with cases B and C, is optimal. Thus, we obtain the largest covering area When linked hexagons, as shown in Figure 2. Obviously, it looks like a honeycomb structure. In fact, the honeycomb pattern is one of the most efficient arrangement for radio spectrum. It transmits by the wireless medium of microwave, satellites and radiation. The structure has a feature of point-to-point transmission or multicast. It is widely used in UN Urban Network, Campus Network and Enterprise Network.Figure 2. some circles intersecting together form the closely linked hexagons Now we have a circle with radius of 40 miles. Then we analyze the distances of signals from users and repeaters covering in the circle. Because the differences for the users and repeaters in energy and height, they have different covering distances. We calculate the distances with the theory of space loss. The formula[6]is1288.120lg 20lg 40lg LM F h h d =+-+,LM the wireless lossF the communication working frequency(MHz)1h the height of the repeater (m)2h the height of the user(m)d the distance between the user and repeater(km) We assume that 150F MHZ =,1 1.5h m = and 230h m =, under the condition of the cable loss and antenna gain, we obtain the system gain()(1,21,2)i j SG Pt PA RA CL RR i j =+-++==.The system gain is the allowed decay maximum of the signal from the users to repeaters. If the system gain value is higher than the wireless loss, the users could communicate with each other. Reversely, the users could not communicate. We make the system gain value equals to the wireless loss, thus, we get the extremity distance between the user and repeater. Then we haveSG LM =We choose a typical repeater and the user facility. Thus, the parameters [6] and data of the repeaters are as followsThe transmitting power 120(43)Pt W dBm =The receiving sensitivity 1116RR dBm =-The antenna gain of the repeaters 9.8RA dB =The cable loss 2CL dB =The parameters of the interphoneThe transmitting power 24(36)Pt W dBm =The receiving sensitivity 2116RR dBm =-The antenna gain of the interphone 0PA dB =The system gain of the system from users to repeaters 1144.2SG dB =. Thus, we get the sending distance from the users to repeaters 113.8d km =. Prove in the same way, we have the system gain of the system from the repeaters to users 2151.2SG dB =, the sending distance from the repeaters to the users 220.7d km =According to the sending distance 113.8D km = between user and repeater as well as the property of regular hexagon, we calculate the distance between two repeaters. We obtain that 223.09D km =, which is described in Figure 3. Because 2D is shorter than 2'D , users in this area cannot communicate with each other. Thus, we consider the sending distance 2'D between two repeaters firstly. Then we calculate the distance between the user and the repeater again shown in Figure 4. Finally, we get that 1'12.4D km =.Figure 3. the calculation distance according to the sending distance from users to repeaters.Figure 4. the calculation distance according to the sending distance from repeaters to the users.According to the calculated distance 12'12.4'21.45D km D km ==, we know that the given circle has a radius of 40 miles. We firstly consider the signals ’ covered area of the repeaters. Thus, we get the distribution of the repeater stations in this area showed in Figure 5. The number of repeater stations is 37. However, we need to decide the amount of repeaters distributing in one station.channel (the signaling channel between two points to transmit and receive signals) to transmit signals. Hence, we need 27 frequency channels [2] to maintain the normal communication.In order to avoid the interference about the close frequency between two repeaters, we arrange each repeater 10 frequency channels. We have121145.0145.03145.06145.09145.6145.63145.66145.69146.2146.23146.26146.29146.8146.83146.86146.89147.4147.43147.46147.49Mhz Mhz MHz MHz Mhz Mhz MHz MHz pl r Mhz Mhz MHz MHz Mhz Mhz MHz MHz Mhz Mhz MHz MHz r ⎧⎧⎪⎪⎪⎪⎪⎪⎨⎨⎪⎪⎪⎪⎩⎩（）233145.12145.15145.72145.75()()146.32146.35146.92146.95147.52147.55MHz MHzMHz MHz pl r pl MHz MHz MHz MHz MHz MHz ⎧⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ Here, n is the number of repeaters.In this method of distribution ，we ensure that the signal could still be recognized after transmission. We associate to each repeater a subaudible tone and the users need to use the same tone to receive the corresponding signal. We suppose each repeater station have the same repeaters attached with different subaudible tones. In this way, we guarantee the signals transmitting in this zone without interference. Because when one user sends a signal with a specific frequency, the repeater could send the signal after adding or subtracting 600 KHz. However, our frequency channels cover the whole scope of the frequency. Thus, the signal can be transmitted in this zone.Finally, we calculate the number of the repeaters in a repeater station and obtain the number is 3. Thus, the total number of the repeaters is 3*37111=.When the number of users in this zone increases to 10000, we consider the problem as the first model. In this situation, each repeater station should cover 10000/37270.3= users. Hence, we need 270 frequency channels to maintain the normal communication. Since the number of the channels is too large, it is wasteful to use 10 frequency channels for the first problem. Thus, we consider assigning each repeater station 30 channels. Furthermore, we get 9 repeaters. However, for the frequency rand ranging from145MHz to 148MHz, the channel changes to 11.1KHz, which leads to the channels interfering with each other. Hence, we make use of the CTCSS system to distribute the 9 repeaters different PL tones. We can build the repeaters which can transmit the same frequency and have different tones.11145145.03145.06145.09145.012145.015145.6145.63145.66145.69145.72145.75()146.2146.23146.26146.29146.32146.35146.8146.83146.86146.89146.92146.95147.4147.43147.46147.49147.52147.55r mhz pl ⎧⎪⎪⎪⎨⎪⎪⎪⎩1'1'145145.03145.06145.09145.012145.015145.6145.63145.66145.69145.72145.75()146.2146.23146.26146.29146.32146.35146.8146.83146.86146.89146.92146.95147.4147.43147.46147.49147.52147.55r mhz pl ⎧⎪⎪⎪⎨⎪⎪⎪⎩Thus, we calculate the number of the repeaters in a repeater station is 270/309=. Then the total number of the repeaters is 9*37333=.The model of the line-of-sight propagation considering the effect ofthe mountainsWe search some information on how to build the repeaters at the top of the mountains. According to the factors influencing the positions of the repeaters, we establish a model to simulate these impact factors of transmission of VHF radio spectrum.When repeaters are installed at the tops of the mountainous, the positions of the repeaters are related to the height of the antenna, its coverage radius, the repeater power and antenna gain. Thus, it is difficult to build the communication network. In order to build communication network well, we should do lots of experiments to ensure the positions of the repeaters according to actual geomorphic environment.Since mountains have different heights, we mainly consider three cases. Case 1 is that the heights of the mountains are 15m below, case 2 requires that the heights ranges from 15 to 30m and the last one is 30m above.The Egli modelThis model considers the height of the mountains below 15m. We assume that the mountains in this zone have no larger peaks, that is, this zone is a medium rolling terrain.This model is based on the data of the mobile communication, which is established by Federal Communications Commission (FCC). It is an empirical equation which is summarized from the data of the irregular terrain. This model based on the barrier height is applied to the VHF radio spectrum and the irregular terrain. It demands the barrier height above 15m. When the barrier height is under 15m ，we modify the model to verify the modified factor T C . The loss of the spectrum [1] equation is218820lg 40lg 20lg 20lg T LM F d h h C =++---.Here, we assume that d is the distance between the two antennas (m), h ∆is the height of thetopography. If we use b h to denote the practical height of the sending signal antenna, o h to denote the least effective height of the antenna and m h the practical height of the receiving signal antenna, then theeffective height of the sending signal antenna 1h satisfies1()2b o h h h m +=, and the effective height of the receiving signal antenna 2h satisfies2()2m o h h h m +=, 100-10-20-301020305070100200300500t h e m o d i f y i n g f a c t o r s K /d B /h mFigure 6[1]. the range of the modifying factor. We obtain the relationship between the height of the topography and the modifying factor from the empirical data. Furthermore, we get the equation with respect to h ∆and T C .C 1.6670.1094h25150T MHz F MHz =-∆<< C 2.250.1476h150162T MHz F MHz =-∆<< C 3.750.2461h 450470T MHz F MHz =-∆<<This model for irregular area is fit for the frequency ranging from 40 to 450MHz. When the frequency is higher than 25MHz or lower than 400MHz and the distance between two antennas is less than 64km, the error would be very small. Through the model we can evaluate the value of the wireless loss and the number of the repeaters.Figure 7 describes the positions of the mobile station, repeater and the barrier. Next, we introduce the concept of the clearance.Figure 7．The schematic of the clearanceT the position of the mobile stationR the position of the repeater1d the distance between the mobile station and the barrier2d the distance between the repeaters and the barrierAssume that the line HD is perpendicular to line RT, which is called clearance showed in Figure 7. Because the distance between the two antennas is very far, thus, the HD is short. Then we can substitute the hd for HC . If the radius of the first Fresnel region (the region is used to evaluate the transmission energy of the video spectrum.) is 1F , we regard 1/HC F as the relative clearance.The equation [2] of the radius of the first Fresnel region is12112d d F d d λ=+where λ is a parameter.When the radio spectrum transmits ，there are always many barriers such as constructions, trees and peaks blocking the spectrum. If the height of the barrier has not reached the first Fresnel zone ，the barrier would have little influence to the receiving frequency level. However, when it is in the zone, it will cause the added losses (the power losses of the sending power relative to the receiving power) to decrease the receiving electrical level. The diffraction losses /dB T h e d i f f r a c t i o n l o s s e s /d BFigure 7. The relationship between diffraction losses and clearance [1].The relationship between the added losses and the clearance caused by the barriers is showed in Figure 7. When the height of the barrier is under the line RT and the relative clearance is larger than 0.5，the added losses changes around 0db. In this situation，the practical receiving electrical level approaches the value of the space loss. We can get the value of the clearance HC is less than0.557F or a negative value. It may1hinder the transmission of direct wave. Thus, we should make the barriers lie below the line RT. Strengths●In the first model, we distribute each repeater 5 frequency channels, meanwhile the different repeatershave different PL tones. Thus, under the condition of avoiding the interference of repeaters with each other, we control the number of frequency channels least to make the transmission more efficient.●The model is established when the users are uniformly distributed. When the number of users increases,the number of repeaters increases. Thus, this model applies the zone where the users are unevenlydistributed.●The Egli model is a model considering the modifying factors, which make the mountains areas problembe easily understood.Weaknesses●In the signal’s coverage area of the repeaters, we assume that each channel only has one user. However,in the practical situation, there may not be one user. That is to say, we have wasted the channel.●Our model belongs to fixed channels distribution strategies, the larger number of the users, the largernumber of the channels. It leads to channel interference with each other when channel bandwidth is less than 8.3MHz. Thus, our model only suits for less number of users.●Considering the mountains environment is complex, in our model, we only consider one mountaineffecting the transmission of radio spectrum.References[1] Yao Dongping, Huang Qing and Zhao Hongli, Digital Microwave Communication, Beijing: Beijing Jiaotong University Press, 2004.7.[2] Theodore S. Rappaport, Wireless Communications: Principles and Practice, Second Edition, Prentice Hall PTR，2006.7[3] DeWitt H.Scott, Michael Krigline, Successful Writing for the Real World, Foreign Language Teaching and Research Press, 2009.2[4] /wiki/Repeater, 2011.2.12[5] /wiki/CTCSS, 2011.2.12[6] /view/2074265.htm,2012.2.14。
美赛历年题目_pdf
马剑整理历年美国大学生数学建模赛题目录MCM85问题-A 动物群体的管理 (3)MCM85问题-B 战购物资储备的管理 (3)MCM86问题-A 水道测量数据 (4)MCM86问题-B 应急设施的位置 (4)MCM87问题-A 盐的存贮 (5)MCM87问题-B 停车场 (5)MCM88问题-A 确定毒品走私船的位置 (5)MCM88问题-B 两辆铁路平板车的装货问题 (6)MCM89问题-A 蠓的分类 (6)MCM89问题-B 飞机排队 (6)MCM90-A 药物在脑内的分布 (6)MCM90问题-B 扫雪问题 (7)MCM91问题-B 通讯网络的极小生成树 (7)MCM 91问题-A 估计水塔的水流量 (7)MCM92问题-A 空中交通控制雷达的功率问题 (7)MCM 92问题-B 应急电力修复系统的修复计划 (7)MCM93问题-A 加速餐厅剩菜堆肥的生成 (8)MCM93问题-B 倒煤台的操作方案 (8)MCM94问题-A 住宅的保温 (9)MCM 94问题-B 计算机网络的最短传输时间 (9)MCM-95问题-A 单一螺旋线 (10)MCM95题-B A1uacha Balaclava学院 (10)MCM96问题-A 噪音场中潜艇的探测 (11)MCM96问题-B 竞赛评判问题 (11)MCM97问题-A Velociraptor(疾走龙属)问题 (11)MCM97问题-B为取得富有成果的讨论怎样搭配与会成员 (12)MCM98问题-A 磁共振成像扫描仪 (12)MCM98问题-B 成绩给分的通胀 (13)MCM99问题-A 大碰撞 (13)MCM99问题-B “非法”聚会 (14)MCM2000问题-A空间交通管制 (14)MCM2000问题-B: 无线电信道分配 (14)MCM2001问题- A: 选择自行车车轮 (15)MCM2001问题-B 逃避飓风怒吼（一场恶风...） .. (15)MCM2001问题-C我们的水系-不确定的前景 (16)MCM2002问题-A风和喷水池 (16)MCM2002问题-B航空公司超员订票 (16)MCM2002问题-C (16)MCM2003问题-A: 特技演员 (18)MCM2003问题-B: Gamma刀治疗方案 (18)MCM2003问题-C航空行李的扫描对策 (19)MCM2004问题-A：指纹是独一无二的吗？ (19)MCM2004问题-B：更快的快通系统 (19)MCM2004问题-C安全与否？ (19)MCM2005问题A.水灾计划 (19)MCM2005B.Tollbooths (19)MCM2005问题C：不可再生的资源 (20)MCM2006问题A: 用于灌溉的自动洒水器的安置和移动调度 (20)MCM2006问题B: 通过机场的轮椅 (20)MCM2006问题C : 抗击艾滋病的协调 (21)MCM2007问题B ：飞机就座问题 (24)MCM2007问题C：器官移植：肾交换问题 (24)MCM2008问题A:给大陆洗个澡 (28)MCM2008问题B：建立数独拼图游戏 (28)MCM85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。
2011数学建模竞赛题目
A: 网络舆论的形成、发展与控制持有、接受、表达某种相同、相似的观点的人在社会人群中所占的比例超过一定的阀值，这时候这种观点就上升为舆论（opinions）。
舆论在特定的条件下，产生巨大的社会力量，能够左右社会大众和政府的行为。
如今，互联网作为一个开放自由的平台，已经成为了世界的“第四媒体”。
显然，网络舆论与传统舆论在形成、发展等方面有着诸多不同的特点，如何控制和引导网络舆论的形成与发展是当今社会的一个重要课题。
作为开放的网络平台，加上其虚拟性、隐蔽性、发散性、渗透性和随意性等特点，越来越多的人们愿意通过互联网来表达自己的个人想法。
现今，互联网已成为新闻集散地、观点集散地和民声集散地。
互联网上的信息内容庞杂多样，容纳了各种人群、各类思潮，对于社会上的一些敏感问题出现在网上而引起一些人的共鸣应是一种正常现象，但是由于各种复杂因素使这些敏感问题向热点演变，最后形成网络舆论并引起社会群众的违规和过激行动时，将影响到社会安定和其他政治问题，因此网络舆论的爆发将以“内容威胁”的形式对社会公共安全形成威胁，对网上的信息内容进行管理和控制将成为互联网进一步发展的必然趋势。
请在上述背景基础上，解决如下问题：（1）请在查找资料的基础上，给出网络舆论的基本概念和特性，分析影响网络舆论的各种因素；（2）运用你们所掌握数学知识，建立网络舆论形成的数学模型，使其能够对网络舆论的发展、变化趋势做出有效的判断，并能对网络舆论的态势做出客观的表述；（3）基于上述模型的基础上，请描述在网络舆论形成后，如何利用你们的模型来网络舆论的发展趋势。
B题：水资源短缺风险综合评价水资源，是指可供人类直接利用，能够不断更新的天然水体。
主要包括陆地上的地表水和地下水。
风险，是指某一特定危险情况发生的可能性和后果的组合。
水资源短缺风险，泛指在特定的时空环境条件下，由于来水和用水两方面存在不确定性，使区域水资源系统发生供水短缺的可能性以及由此产生的损失。
2011 AIME I中文版
2011年美国数学邀请赛(1)1.瓶子A中有4升45%的酸溶液,瓶子B中有5升48%的酸溶液,瓶m升溶液添加到瓶子A 子C中有1升k%的酸溶液,将瓶子C中的n中,并将瓶子C中剩余的溶液都添加到瓶子B中,结束后,瓶子A和瓶子B中都是50%的酸溶液.已知m和n是互质的正整数,求k+m+n.2.在矩形ABCD中,AB=12,BC=10,点E和F在矩形ABCD的内部,使得BE=9,DF=8,BE//DF,EF//AB,直线BE与线段AD相交.EF的长度可以表示成m n-p,这里m,n,p是正整数,且n不能被任何质数的平方整除,求m+n+p.5的直线,令M是过点B(5,6),并且与3.令L过点A(24,-1),且斜率为12L垂直的直线.抹去原来的坐标系,以直线L为x轴,直线M为y轴,在新的坐标系中,点A在x轴的正半轴上,点B在y轴的正半轴上,原坐标系中坐标为(-14,27)的点P在新坐标系中的坐标是(α,β),求α+β.4.在∆ABC中,AB=125,AC=117,BC=120,∠A的平分线交BC于点L,∠B的平分线交AC于点K,过C作BK和AL的垂线,垂足分别是M和N,求MN.5.将1~9这九个数字标在一个正九边形的顶点上,使每三个连续顶点上的数字之和是3的倍数.如果一个满足要求的排列可由另一个排列经过九边形在平面上的旋转而得到,则认为它们是相同的.求所有不相同的排列的个数.6. 设方程是y=ax 2+bx+c 的抛物线的顶点是(41,-89),这里a>0,a+b+c 是一个整数,a 的最小可能的取值可写成qp 的形式,这里p,q 是互质的正整数,求p+q.7. 若存在非负整数x 0,x 1,⋯,x 2011,使得0x m =∑=20111k x k m ,其中m 是正整数,求这样的m 的个数.8. 在∆ABC 中,BC=23,CA=27,AB=30.点V 和W 在AC 上,且V 在AW 上,点X 和Y 在BC 上,且X 在CY 上,点Z 和U 在AB 上,且Z 在UB 上,这些点使得UV//BC,WX//AB,YZ//CA.沿着UV,WX,YZ 折叠,使得两面成直角.图示的结果是一张放在水平面上有三角形腿的桌子,h 是由三角形ABC 构作的桌面与地面平行的桌子的最大高度,h 可以表示成n mk 的形式,这里k 和n 是互质的正整数,m 是不能被任何质数的平方整除的正整数,求k+m+n.9. 设x ∈[0,2π],且log 24sinx (24cosx)=23,求24cot 2x. 10. 从一个正n 边形的顶点中随机地选取三个顶点形成钝角三角形的概率是12593,求所有可能的n 的值之和.11. 形如2n (n 是非负整数)的数被1000除,所有可能的余数形成集合R,令S是R中元素之和,求S被1000除的余数.12.六名男子和若干名女子按随机的顺序排成一列,当每名男子的边上至少有另一名男子时,至少有一组四名男子站在一起的概率是p,求使得p不超过1%的女子的人数的最小值.13.一个棱长为10的正方体悬挂在平面的上方,离平面最近的顶点记作A,与顶点A相邻的三个顶点在平面上方的高度是10,11,12,顶点A到平面的距离可以表示成t sr+,这里r,s,t是正整数,求r+s+t. 14.设A1A2A3A4A5A6A7A8是一个正八边形,M1,M3,M5,M7分别是A1A2,A3A4,A5A6,A7A8的中点,对i=1,3,5,7,射线R从M i并射向八边形的内部,使得R1⊥R3,R3⊥R5,R5⊥R7,R7⊥R1,射线R1与R3,R3与R5,R5与R7,R7与R1分别相交于B1,B3,B5,B7,如果A1A2=B1B3,则cos2∠A3M3B1可以写成m-n的形式,这里m和n是正整数,求m+n.15.有一些整数m,使得多项式x3-2011x+m有三个整数根a,b,c,求|a|+|b|+|c|.答案085 036 031 056 144011 016 318 192 503007 594 330 037 098。
2011美国数学竞赛(A和B)
2011A1A cell phone plan costs$20dollars each month,plus5cents per text message sent,plus10 cents for each minute used over30hours.In January Michelle sent100text messages and talked for30.5hours.How much did she have to pay?(A)$24.00(B)$24.50(C)$25.50(D)$28.00(E)$30.002There are5coins placedflat on a table according to thefigure.What is the order of the coins from top to bottom?(A)(C,A,E,D,B)(B)(C,A,D,E,B)(C)(C,D,E,A,B)(D)(C,E,A,D,B)(E)(C,E,D,A,B)ABCDE3A small bottle of shampoo can hold35milliliters of shampoo,whereas a large bottle can hold500milliliters of shampoo.Jasmine wants to buy the minimum number of small bottles necessary to completelyfill a large bottle.How many bottles must she buy?(A)11(B)12(C)13(D)14(E)154At an elementary school,the students in third grade,fourth grade,andfifth grade run an average of12,15,and10minutes per day,respectively.There are twice as many third graders as fourth graders,and twice as many fourth graders asfifth graders.What is the average number of minutes run per day by these students?(A)12(B)373(C)887(D)13(E)145Last summer30%of the birds living on Town Lake were geese,25%were swans,10%were herons,and35%were ducks.What percent of the birds that were not swans were geese?(A)20(B)30(C)40(D)50(E)60Thisfile was downloaded from the AoPS Math Olympiad Resources Page Page120116The players on a basketball team made some three-point shots,some two-point shots,and some one-point free throws.They scored as many points with two-point shots as with three-point shots.Their number of successful free throws was one more than their number of successful two-point shots.The team’s total score was61points.How many free throws did they make?(A)13(B)14(C)15(D)16(E)177A majority of the30students in Ms.Demeanor’s class bought penciles at the school bookstore.Each of these students bought the same number of pencils,and this number was greater than1.The cost of a pencil in cents was greater than the number of pencils each student bought,and the total cost of all the pencils was$17.71.What was the cost of a pencil in cents?(A)7(B)11(C)17(D)23(E)778In the eight-term sequence A,B,C,D,E,F,G,H,the value of C is5and the sum of any three consecutive terms is30.What is A+H?(A)17(B)18(C)25(D)26(E)439At a twins and triplets convention,there were9sets of twins and6sets of triplets,all from different families.Each twin shook hands with all the twins except his/her sibling and with half the triplets.Each triplet shook hands with all the triplets except his/her siblings and half the twins.How many handshakes took place?(A)324(B)441(C)630(D)648(E)88210A pair of standard6-sided fair dice is rolled once.The sum of the numbers rolled determines the diameter of a circle.What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle’s circumference?(A)136(B)112(C)16(D)14(E)51811Circles A,B,and C each have radius1.Circles A and B share one point of tangency.CircleC has a point of tangency with the midpoint of AB.What is the area inside circle C butoutside circle A and circle B?(A)3−π2(B)π2(C)2(D)3π4(E)1+π212A power boat and a raft both left dock A on a river and headed downstream.The raft drifted at the speed of the river current.The power boat maintained a constant speed with respect to the river.The power boat reached dock B downriver,then immediately turned and traveled back upriver.It eventually met the raft on the river9hours after leaving dock A.How many hours did it take the power boat to go from A to B?(A)3(B)3.5(C)4(D)4.5(E)5201113Triangle ABC has side-lengths AB =12,BC =24,and AC =18.The line through theincenter of ABC parallel to BC intersects AB at M and AC at N .What is the perimeter of AMN ?(A)27(B)30(C)33(D)36(E)4214Suppose a and b are single-digit positive integers chosen independently and at random.Whatis the probability that the point (a,b )lies above the parabola y =ax 2−bx ?(A)1181(B)1381(C)527(D)1781(E)198115The circular base of a hemisphere of radius 2rests on the base of a square pyramid of height6.The hemisphere is tangent to the other four faces of the pyramid.What is the edge-length of the base of the pyramid?(A)3√2(B)133(C)4√2(D)6(E)13216Each vertex of convex pentagon ABCDE is to be assigned a color.There are 6colors tochoose from,and the ends of each diagonal must have different colors.How many different colorings are possible?(A)2520(B)2880(C)3120(D)3250(E)375017Circles with radii 1,2,and 3are mutually externally tangent.What is the area of the triangledetermined by the points of tangency?(A)35(B)45(C)1(D)65(E)4318Suppose that |x +y |+|x −y |=2.What is the maximum possible value of x 2−6x +y 2?(A)5(B)6(C)7(D)8(E)919At a competition with N players,the number of players given elite status is equal to21+ log 2(N −1) −N.Suppose that 19players are given elite status.What is the sum of the two smallest possible values of N ?(A)38(B)90(C)154(D)406(E)102420Let f (x )=ax 2+bx +c ,where a ,b ,and c are integers.Suppose that f (1)=0,50<f (7)<60,70<f (8)<80,and 5000k <f (100)<5000(k +1)for some integer k .What is k ?(A)1(B)2(C)3(D)4(E)521Let f 1(x )=√1−x ,and for integers n ≥2,let f n (x )=f n −1(√n 2−x ).If N is the largestvalue of n for which the domain of f n is nonempty,the domain of f N is c .What is N +c ?(A)−226(B)−144(C)−20(D)20(E)144201122Let R be a square region and n ≥4an integer.A point X in the interior of R is called n-raypartitional if there are n rays emanating from X that divide R into n triangles of equal area.How many points are 100-ray partitional but not 60-ray partitional?(A)1500(B)1560(C)2320(D)2480(E)250023Let f (z )=z +a z +b and g (z )=f (f (z )),where a and b are complex numbers.Suppose that|a |=1and g (g (z ))=z for all z for which g (g (z ))is defined.What is the difference between the largest and smallest possible values of |b |?(A)0(B)√2−1(C)√3−1(D)1(E)224Consider all quadrilaterals ABCD such that AB =14,BC =9,CD =7,DA =12.What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?(A)√15(B)√21(C)2√6(D)5(E)2√725Triangle ABC has ∠BAC =60◦,∠CBA ≤90◦,BC =1,and AC ≥AB .Let H ,I ,and Obe the orthocenter,incenter,and circumcenter of ABC ,respectively.Assume that the area of the pentagon BCOIH is the maximum possible.What is ∠CBA ?(A)60◦(B)72◦(C)75◦(D)80◦(E)90◦2011B1What is2+4+6 1+3+5−1+3+52+4+6?(A)−1(B)536(C)712(D)14760(E)4332Josanna’s test scores to date are90,80,70,60,and85.Her goal is to raise her test average at least3points with her next test.What is the minimum test score she would need to accomplish this goal?(A)80(B)82(C)85(D)90(E)953LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally.Over the week,each of them paid for various joint expenses such as gasoline and car rental.At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paidB dollars,where A<B.How many dollars must LeRoy give to Bernardo so that they sharethe costs equally?(A)A+B2(B)A−B2(C)B−A2(D)B−A(E)A+B4In multiplying two positive integers a and b,Ron reversed the digits of the two-digit numbera.His errorneous product was161.What is the correct value of the product of a and b?(A)116(B)161(C)204(D)214(E)2245Let N be the second smallest positive integer that is divisible by every positive integer less than7.What is the sum of the digits of N?(A)3(B)4(C)5(D)6(E)96Two tangents to a circle are drawn from a point A.The points of contact B and C divide the circle into arcs with lengths in the ratio2:3.What is the degree measure of∠BAC?(A)24(B)30(C)36(D)48(E)607Let x and y be two-digit positive integers with mean60.What is the maximum value of the ratio xy?(A)3(B)337(C)397(D)9(E)99108Keiko walks once around a track at exactly the same constant speed every day.The sides of the track are straight,and the ends are semicircles.The track has width6meters,and2011it takes her36seconds longer to walk around the outside edge of the track than around the inside edge.What is Keiko’s speed in meters per second?(A)π3(B)2π3(C)π(D)4π3(E)5π39Two real numbers are selected independently at random from the interval[-20,10].What is the probability that the product of those numbers is greater than zero?(A)19(B)13(C)49(D)59(E)2310Rectangle ABCD has AB=6and BC=3.Point M is chosen on side AB so that∠AMD=∠CMD.What is the degree measure of∠AMD?(A)15(B)30(C)45(D)60(E)7511A frog located at(x,y),with both x and y integers,makes successive jumps of length5and always lands on points with integer coordinates.Suppose that the frog starts at(0,0)and ends at(1,0).What is the smallest possible number of jumps the frog makes?(A)2(B)3(C)4(D)5(E)612A dart board is a regular octagon divided into regions as shown.Suppose that a dart thrown at the board is equally likely to land anywhere on the board.What is probability that the dart lands within the center square?(A)√2−12(B)14(C)2−√22(D)√24(E)2−√213Brian writes down four integers w>x>y>z whose sum is44.The pairwise positive differences of these numbers are1,3,4,5,6,and9.What is the sum of the possible values for w?(A)16(B)31(C)48(D)62(E)93201114A segment through the focus F of a parabola with vertex V is perpendicular to F V andintersects the parabola in points A and B .What is cos(∠AV B )?(A)−3√57(B)−2√55(C)−45(D)−35(E)−1215How many positive two-digit integers are factors of 224−1?(A)4(B)8(C)10(D)12(E)1416Rhombus ABCD has side length 2and ∠B =120◦.Region R consists of all points inside therhombus that are closer to vertex B than any of the other three vertices.What is the area of R ?(A)√33(B)√32(C)2√33(D)1+√33(E)217Let f (x )=1010x ,g (x )=log 10 x 10,h 1(x )=g (f (x )),and h n (x )=h 1(h n −1(x ))for inte-gers n ≥2.What is the sum of the digits of h 2011(1)?(A)16,081(B)16,089(C)18,089(D)18,098(E)18,09918A pyramid has a square base with sides of length 1and has lateral faces that are equilateraltriangles.A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid.What is the volume of this cube?(A)5√2−7(B)7−4√3(C)2√227(D)√29(E)√3919A lattice point in an xy -coordinate system is any point (x,y )where both x and y are integers.The graph of y =mx +2passes through no lattice point with 0<x ≤100for all m such that 12<m <a .What is the maximum possible value of a ?(A)51101(B)5099(C)51100(D)52101(E)132520Triangle ABC has AB =13,BC =14,and AC =15.The points D,E,and F are the mid-points of AB ,BC ,and AC respectively.Let X =E be the intersection of the circumcircles of BDE and CEF .What is XA +XB +XC ?(A)24(B)14√3(C)1958(D)129√714(E)69√2421The arithmetic mean of two distinct positive integers x and y is a two-digit integer.Thegeometric mean of x and y is obtained by reversing the digits of the arithmetic mean.What is |x −y |?(A)24(B)48(C)54(D)66(E)7022Let T 1be a triangle with sides 2011,2012,and 2013.For n ≥1,if T n = ABC and D,E,and F are the points of tangency of the incircle of ABC to the sides AB,BC and AC ,2011respectively,then T n +1is a triangle with side lengths AD,BE,and CF ,if it exists.What is the perimeter of the last triangle in the sequence (T n )?(A)15098(B)150932(C)150964(D)1509128(E)150925623A bug travels in the coordinate plane,moving only along the lines that are parallel to thex-axis or y-axis.Let A =(−3,2)and B =(3,−2).Consider all possible paths of the bug from A to B of length at most 20.How many points with integer coordinates lie on at least one of these paths?(A)161(B)185(C)195(D)227(E)25524Let P (z )=z 8+(4√3+6)z 4−(4√3+7).What is the minimum perimeter among all the8-sided polygons in the complex plane whose vertices are precisely the zeros of P (z )?(A)4√3+4(B)8√2(C)3√2+3√6(D)4√2+4√3(E)4√3+625For every m and k integers with k odd,denote by [m k ]the integer closest to m k .For every odd integer k ,let P (k )be the probability that[n k ]+[100−n k ]=[100k]for an integer n randomly chosen from the interval 1≤n ≤99!.What is the minimum possible value of P (k )over the odd integers K in the interval 1≤k ≤99?(A)12(B)5099(C)4487(D)3467(E)713。
2011年美赛真题优秀论文
中继站的协调方案摘要（Abstract ）中继站是将信号进行再生、放大处理后，再转发给下一个中继站，以确保传输信号的质量。
低功耗的用户，例如移动电话用户，在不能直接与其他用户联系的地方可以通过中继站来保持联系。
然而,中继站之间会互相影响,除非彼此之间有足够远的距离或通过充分分离的频率来传送。
为了排除信号间的干扰，实现某一区域内（题中以40英里为半径的圆形区域）通信设备正常的发射和接收信号，需要利用PL 技术对中继站作合理的协调和分配。
首先本文结合香农理论的相关算法，考虑了信号供给系统的损耗、天线增益、信号的传播损耗、辐射效率因素的影响，得到中继站的辐射范围半径公式为：,10,10log ()37.23282010r outr inp P d -=在供给对象为低功率消耗设备，查资料一般发射功率为3.2W ，中继站能接收到的最弱的信号1W μ,代入数据得到每个中继站的辐射半径为15.28m iles 。
同时本文在不考虑其他因素（包括：地形、大雾、山川、建筑物等）对辐射范围和辐射强度的影响下，结合相关知识和题目中给出的条件，在不引入PL 技术时得出每个中继站所服务的用户数量为39个。
对于问题一， 我们首先定义了均衡覆盖、覆盖效率，在均衡覆盖中即用圆覆盖圆形区域，我们根据式子2(2)n k n ππ-=，得出(,)k n 的可能值有(3,6),(4,4),三种，即等效三角形、正方形、正六边形覆盖，并通过覆盖效率的比较，最终得出正六边形覆盖是最好的覆盖方法，即蜂窝拓扑网络。
在这种覆盖情况下我们，我结合中继站覆盖半径15.28m iles ，根据式子m i n 3(1)1,0,1,2,3,N K K K =++=……，求出最少需要19个中继站，并在满足单位面积覆盖同时在线人数的情况下引入PL 技术，得出此时中继站在该区域可同时服务在限人数为1292人。
对于问题二，我们在问题一模型基础上从提高中继站服务人数和减少中继站半径两方面考虑，得出在将PL 分为18层，即中继站同时在线服务人数为702的情况下，结合单位面积同时在线服务人数，得出在中继站最少的情况下，中继站半径在[]11.094，,11.68范围内都可，我们为了让同时在线服务人数最大，取11.094英里，得出服务人数为11305。
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2002年美国大学生数学建模竞赛题目
2002 Mathematical Contest in Modeling （MCM）Problems
问题A
作者：Tjalling Ypma
标题：风和喷水池
在一个楼群环绕的宽阔的露天广场上，装饰喷泉把水喷向高空。
刮风的日子，风把水花从喷泉吹向过路行人。
喷泉射出的水流受到一个与风速计（用于测量风的速度和方向）相连的机械装置控制，前者安装在一幢邻近楼房的顶上。
这个控制的实际目标，是要为行人在赏心悦目的景象和淋水浸湿之间提供可以接受的平衡：风刮得越猛，水量和喷射高度就越低，从而较少的水花落在水池范围以外。
你的任务是设计一个算法，随着风力条件的变化，运用风速计给出的数据来调整由喷泉射出的水流。
Problem A
Authors: Tjalling Ypma
Title: Wind and Waterspray
An 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.
问题B
作者：Bill Fox 和 Rich West
标题：航空公司超员订票
你备好行装准备去旅行，访问New York城的一位挚友。
在检票处登记之后，航空公司职员告诉说，你的航班已经超员订票。
乘客们应当马上登记以便确定他们是否还有一个座位。
航空公司一向清楚，预订一个特定航班的乘客们只有一定的百分比将实际乘坐那个航班。
因而，大多数航空公司超员订票?也就是，他们办理超过飞机定员的订票手续。
而有时，
1
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