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。
2011MCM美赛A题论文 repeater coordination
2 General observations
In this section we discuss observations of the problem statement, ambiguities, and general assumptions we have made to simplify and clarify the modeling process. Additional clarification of some assumptions is provided in the appendices.
Optimal Placement of Radio Repeater Networks
Control #10754 February 14, 2011
Abstract In this paper we consider the problem of placing radio repeaters to serve users in an area. Given a population distribution and geographical map, we use a hillclimbing algorithm to find a minimum number of repeaters required to cover an area and then a genetic algorithm to provide maximal population coverage and network connectedness. We then use hill-climbing techniques to allocate subnetworks based on population size at repeater locations so that two arbitrary users can communicate even when all other users are communicating over the maximum number of possible networks. Our resulting algorithm is capable of producing a range of repeater network allocations, from robust networks that are capable of handling worstcase usage scenarios to smaller networks that provide optimal population coverage and connectivity. On a set of real-world population and geography data, we found that the combination of the hill-climbing and genetic algorithms had 28% better population coverage than a control algorithm did, as well as higher connectivity.
11年数学建模美国赛A题
11A 设计单板滑雪场摘要本文研究的是单板滑雪轨道的设计问题。
首先，我们考虑使运动员的腾空 高度尽量达到最大，建立以腾空高度为目标函数的模型， 针对问题一， 要使滑雪者的垂直距离最长，通过对滑雪者运动过程的分析， 我们选取滑雪者运动的一个周期进行研究， P 平台通过 F 平台再到到 P 平台的 从 另一端，即从 A—B—C—D(如图 2)。
首先， 结合给定的滑雪路线 A—B—C—D 运用物理学知识对滑雪过程进行分 析，建立以最大腾空高度为目标函数，滑雪道曲率半径 r 、倾斜角 、底部平台 宽度 d 以及运动员出槽时与槽边缘的夹角 为自变量的微分方程， 根据搜索相关 资料得知，滑雪道长度、曲率半径、倾斜角及底部平台宽度的设计都有一定的参 考范围限制， 我们结合 U 型单板滑雪道设计数据。
其次， 对于滑雪道长度的求解， 我们是根据运动员的平均腾空次数及每次腾空时所行距离 l 求解得出，而腾空 次数的确定是取 5 次腾空为标准。
关键词： 受力分析 能量守恒 微分方程一、问题重述单板滑雪场地主要由 Flat 平台、过渡区、垂直区、Platform 平台、入口坡组 成。
运动员的滑雪技巧、身体素质，滑雪场地的坡度、深度、宽度、长度等成为 影响运动员成绩的诸多因素。
在单板滑雪比赛中，当滑雪运动员最大限度地产生 垂直腾空后就能够做出各种动作，那么应该如何考虑哪些权衡因素，从而设计出 比较优化的单板滑雪场地。
确定一个滑雪场的形状，使得滑雪选手垂直腾空高度最大化。
“垂直腾空” 最大化就是指最大的垂直腾空距离在半管边缘以上的距离。
定制形状时要优化其他可能的要求， 如空气中的最大扭曲等。
综合各种条件， 选择最优的滑雪场模型。
初步观察得到U型管道的合理设计有利于运动员水平的发挥，因此我们从设 计U型槽的坡度，弧长，场地长度等角度入手。
需要建立数学模型解决的问题： （1）设计出滑雪道的形状（现在一般为半圆柱管内侧面） ，以使得滑雪者的垂 直距离（滞空时间）最长。
2009-2011年美赛题目(1)
2009 Contest ProblemsMCM PROBLEMSPROBLEM A: Designing a Traffic CircleMany cities and communities have traffic circles—from large ones with many lanes in the circle (such as at the Arc de Triomphe in Paris and the Victory Monument in Bangkok) to small ones with one or two lanes in the circle. Some of these traffic circles position a stop sign or a yield sign on every incoming road that gives priority to traffic already in the circle; some position a yield sign in the circle at each incoming road to give priority to incoming traffic; and some position a traffic light on each incoming road (with no right turn allowed on a red light). Other designs may also be possible.The goal of this problem is to use a model to determine how best to control traffic flow in, around, and out of a circle. State clearly the objective(s) you use in your model for making the optimal choice as well as the factors that affect this choice. Include a Technical Summary of not more than two double-spaced pages that explains to a Traffic Engineer how to use your model to help choose the appropriate flow-control method for any specific traffic circle. That is, summarize the conditions under which each type of traffic-control method should be used. When traffic lights are recommended, explain a method for determining how many seconds each light should remain green (which may vary according to the time of day and other factors). Illustrate how your model works with specific examples.PROBLEM B: Energy and the Cell PhoneThis quest ion involves the “energy” consequences of the cell phone revolution. Cell phone usage is mushrooming, and many people are using cell phones and giving up their landline telephones. What is the consequence of this in terms of electricity use? Every cell phone comes with a battery and a recharger.Requirement 1Consider the current US, a country of about 300 million people. Estimate from available data the number H of households, with m members each, that in the past were serviced by landlines. Now, suppose that all the landlines are replaced by cell phones; that is, each of the m members of the household has a cell phone. Model the consequences of this change for electricity utilization in the current US, both during the transition and during the steady state. The analysis should take into account the need for charging the batteries of the cell phones, as well as the fact that cell phones do not last as long as landline phones (for example, the cell phones get lost and break).Requirement 2Consider a se cond “Pseudo US”—a country of about 300 million people with about the sameeconomic status as the current US. However, this emerging country has neither landlines nor cell phones. What is the optimal way of providing phone service to this country from an energy perspective? Of course, cell phones have many social consequences and uses that landline phones do not allow. A discussion of the broad and hidden consequences of having only landlines, only cell phones, or a mixture of the two is welcomed.Requirement 3Cell phones periodically need to be recharged. However, many people always keep their recharger plugged in. Additionally, many people charge their phones every night, whether they need to be recharged or not. Model the energy costs of this wasteful practice for a Pseudo US based upon your answer to Requirement 2. Assume that the Pseudo US supplies electricity from oil. Interpret your results in terms of barrels of oil.Requirement 4Estimates vary on the amount of energy that is used by various recharger types (TV, DVR, computer peripherals, and so forth) when left plugged in but not charging the device. Use accurate data to model the energy wasted by the current US in terms of barrels of oil per day.Requirement 5Now consider population and economic growth over the next 50 years. How might a typical Pseudo US grow? For each 10 years for the next 50 years, predict the energy needs for providing phone service based upon your analysis in the first three requirements. Again, assume electricity is provided from oil. Interpret your predictions in term of barrels of oil.MCM2009问题A : 设计一个交通环岛在许多城市和社区都建立有交通环岛，既有多条行车道的大型环岛（例如巴黎的凯旋门和曼谷的胜利纪念碑路口），又有一至两条行车道的小型环岛。
美国大学生数学建模竞赛试题_ICM_Problem_D
As you develop your model and prepare to test it, you will need to assemble a collection of data. Below are just some examples of the types of data you may find useful in this project. Depending on your exact model, some types of data may be very important and others may be entirely irrelevant. In addition to the sample sources provided below, you might want to consider a few important world events throughout history – if some recent big news events, such as the rumors of country-turned-pop singer Taylor Swift’s possible engagement had instead happened in 1860, what percentage of the population would know about it and how quickly; likewise, if an important person was assassinated today, how would that news spread? How might that compare to the news of US President Abraham Lincoln’s assassination?
2011年美国大学生数学建模竞赛培训课件
2011年美国大学生数学建模竞赛培训课件内容：2010年研究生数学建模竞赛D题题目特殊工件磨削加工的数学建模某科研单位和工厂研制了一种大型精密内外圆曲线磨床，用来加工具有复杂母线旋转体的特殊工件，如导弹天线罩等，这些工件具有硬度高、尺度大、加工精度高和母线为连续光滑曲线等特点。
图1是几类加工工件示例，工件1的内外母线均为凸的，工件2的内母线是非单调凸的。
这些工件的最后精密成形工艺采用磨削加工。
图1 几类特殊加工工件示例该磨床主要由机床底座，下工作台，中工作台，上工作台（简称下台、中台和上台），工件工作箱和砂轮机箱等组成（见图2，其中仅画出砂轮而未显示砂轮机箱）。
下台、中台可分别沿着设在底座和下台上的直导轨作直线运动，这两组导轨相互垂直；上台能沿中台上的圆导轨作转动。
驱动砂轮高速旋转的砂轮机箱安装在机床底座上，砂轮的旋转轴线与底座导轨方向保持平行，且与工件工作箱的旋转主轴等高（即两旋转轴线位于同一水平面）。
各工作台的移动量均可在机床控制面板上自动显示。
图2所示为磨削工件外表面时的情况，更换砂轮后可加工内表面。
图2 大型数控精密内外圆磨床的结构示意图工件工作箱固装在上台上，它通过专用夹具装夹工件，使工件绕工件工作箱主轴以较慢的转速旋转，同时随三个工作台的复合运动改变待加工工件与砂轮的相对位置。
三个台的运动必须相互配合，使工件与砂轮相切磨削，加工出满足要求的旋转体。
三个工作台的运动分别由三组步进电机控制。
步进电机是一种精密数控电动机，每输入一个控制脉冲，电机主轴转动一个精确的步进角度（正向或反向），它的大小与方向由电机结构和控制电路确定（改变电机诸绕组的通电顺序就可改变其旋转方向）；既可输入适当个数的脉冲控制电机主轴的角位移量；也可通过控制某时段中的脉冲频率或脉冲的分布使电机主轴转动速度达到某种要求：若某时段中的脉冲频率为常数（即脉冲为均匀分布），则电机主轴的转动可视为匀速，否则为变速，从而实现调速。
2011年美国大学生数学建模大赛赛题
2011 MCM ProblemsPROBLEM A: Snowboard CourseDetermine the shape of a snowboard course (currently known as a “halfpipe”) to maximize the production of “vertical air” by a skilled snowboarder. "Vertical air" is the maximum vertical distance above the edge of the halfpipe. Tailor the shape to optimize other possible requirements, such as maximum twist in the air.What tradeoffs may be required to develop a “practical” course?PROBLEM B: Repeater CoordinationThe VHF radio spectrum involves line-of-sight transmission and reception. This limitation can be overcome by “repeaters,” which pick up weak signals, amplify them, and retransmit them on a different frequency. Thus, using a repeater, low-power users (such as mobile stations) can communicate with one another in situations where direct user-to-user contact would not be possible. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.In addition to geographical sepa ration, the “continuous tone-coded squelch system” (CTCSS), sometimes nicknamed “private line” (PL), technology can be used to mitigate interference problems. This system associates to each repeater a separate subaudible tone that is transmitted by all users who wish to communicate through that repeater. The repeater responds only to received signals with its specific PL tone. With this system, two nearby repeaters can share the same frequency pair (for receive and transmit); so more repeaters (and hence more users) can be accommodated in a particular area.For a circular flat area of radius 40 miles radius, determine the minimum number of repeaters necessary to accommodate 1,000 simultaneous users. Assume that the spectrum available is 145 to 148 MHz, the transmitter frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.How does your solution change if there are 10,000 users?Discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas.。
2011年美国数学建模C题汉语版
电动汽车作为一个普遍的手段交通Rick BaileyBrenda HowellZachary StankoHumboldt State UniversityArcata, CAAdvisor: Brad Finney摘要我们适应一个Lotka-Volterra生态竞争模型来描述汽车(和轻型卡车)市场。
我们假设汽油内部内燃机车辆(ICE),插电式混合动力车(PHEV),和电池动力汽车杆状执行像生物竞争一个共享的但有限的资源。
对于生物,这个资源可能是一个食品供应;在汽车市场,制造商争夺消费者的钱。
这个方程描述利率变化的三个因变量,每种类型的汽车的数量。
该模型参数描述增长利率,种间竞争,和承受能力,也间接的联系对消费者偏好、经济条件下,政府的影响,在汽车技术和改进。
变量和参数模型中使用的表1中列出。
我们假设内在增长速率常数,但兼容模型可以描述它们作为函数的时候,市场力量,或随机变量。
我们假设承载能力以1%的速度增长,一致的与人类的人口增长率为美国[世界银行集团2011]。
我们将一起模型参数,以确定的变量反映各方面影响消费者的选择。
我们调查的5个场景变化影响汽车市场。
一个基本场景使用当前的年度增长率和当前的人口;其他调查高油价的影响,提高电池的性能。
政府投资和高电价。
UMAP杂志的32(2)(2011)165 - 178。
c 2011年版权届时系统公司。
保留所有权利。
许可,将数字或硬拷贝的部分或全部这项工作为个人或课堂使用授予没有费只要副本没有制造或分布式的利润或商业优势,此通知副本熊。
抽象与信贷是允许的,但版权组件的这项工作由其他人拥有比届时系统必须遵守的。
复制否则,全文转载,发布服务器上,或重新分配到列表需要事先同意届时系统。
我们比较了两种车型的现值目前可用的,来检查这些汽车的竞争力。
没有目前的政府补贴,尼桑Leaf有较低的现值比本田思域和因此处于不利地位对公民。
Leaf将竞争没有补贴与线性上升在天然气价格5美元/加仑,增加数十亿效率(kWh /英里驱动),和更高的转售价值。
2011年美国大学生数学竞赛试题
1 1 + (x − ai )2
2
dx ≤ An.
Prove there is a constant B > 0 such that for all n,
n i,j =1
(1 + (ai − aj )2 ) ≥ Bn3 .
be a (not necessarily minimal) set of distinct generators of G. A special die, which randomly selects one
m→∞
lim
1 b2m
x∈G
Prob(g = x) −
1 n
2
is positive and finite. B1 Let h and k be positive integers. Prove that for every ǫ > 0, there are positive integers m and n such that √ √ ǫ < |h m − k n| < 2ǫ. B2 Let S be the set of all ordered triples (p, q, r) of prime numbers for which at least one rational number x satisfies px2 + qx + r = 0. Which primes appear in seven or more elements of S ? B3 Let f and g be (real-valued) functions defined on an open interval containing 0, with g nonzero and continuous at 0. If f g and f /g are differentiable at 0, must f be differentiable at 0? B4 In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two 2011 × 2011 matrices, T = (Thk ) and W = (Whk ). Initially, T = W = 0. After every game, for every (h, k ) (including for h = k ), if players h and k tied (that is, both won or both lost), the entry Thk is increased by 1, while if player h won and player k lost, the entry Whk is increased by 1 and Wkh is decreased by 1. Prove that at the end of the tournament, det(T + iW ) is a non-negative integer divisible by 22010 . B5 Let a1 , a2 , . . . be real numbers. Suppose that there is a constant A such that for av ) ∈ R2 , the vector ∇F (u, v ) is either 0 or parallel to the vector g (u), −g (v ) . Prove that there exists a constant C such that for every n ≥ 2 and any x1 , . . . , xn+1 ∈ R, we have C min |F (xi , xj )| ≤ . i=j n A6 Let G be an abelian group with n elements, and let {g1 = e, g2 , . . . , gk } G
2011年美国大学生数学建模竞赛Meritorious
2.3.1 Uniform Distribution for Users...............................................................5 2.3.2 Nonuniform Distribution for Users.........................................................7 2.4 Solution of Model (For 10,000 users)...............................................................8 2.4.1 SchemeⅠ................................................................................................8 2.4.3 SchemeⅡ................................................................................................9 2.4.3 Comparison of Two Schemes ...............................................................10 2.5 Sensitivity Analyses ........................................................................................10
2000-2013美国数学建模竞赛(MCM、ICM)历年试题汇总
目录2000 年美国大学生数学建模竞赛MCM、ICM 试题 (3)2000 MCM A: Air Traffic Control (3)2000 MCM B: Radio Channel Assignments (3)2000 ICM: Elephants: When is Enough, Enough? (5)2001 年美国大学生数学建模竞赛MCM、ICM 试题 (7)2001 MCM A: Choosing a Bicycle Wheel (7)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...). (8)2001 ICM: Our Waterways - An Uncertain Future (10)2002 年美国大学生数学建模竞赛MCM、ICM 试题 (14)2002 MCM A: Wind and Waterspray (14)2002 MCM B: Airline Overbooking (14)2002 ICM: Scrub Lizards (15)2003 年美国大学生数学建模竞赛MCM、ICM 试题 (19)2003 MCM A: The Stunt Person (19)2003 MCM B: Gamma Knife Treatment Planning (19)2003 ICM: Aviation Baggage Screening Strategies: To Screen or Not to Screen, that is the Question (20)2004 年美国大学生数学建模竞赛MCM、ICM 试题 (24)2004 MCM A: Are Fingerprints Unique? (24)2004 MCM B: A Faster QuickPass System (24)2004 ICM: To Be Secure or Not to Be? (24)2005 年美国大学生数学建模竞赛MCM、ICM 试题 (25)2005 MCM A: Flood Planning (25)2005 MCM B: Tollbooths (25)2005 ICM: Nonrenewable Resources (25)2006 年美国大学生数学建模竞赛MCM、ICM 试题 (27)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation (27)2006 MCM B: Wheel Chair Access at Airports (27)2006 ICM: Trade-offs in the fight against HIV/AIDS (28)2007 年美国大学生数学建模竞赛MCM、ICM 试题 (32)2007 MCM A: Gerrymandering (32)2007 MCM B: The Airplane Seating Problem (32)2007 ICM: Organ Transplant: The Kidney Exchange Problem (33)2008 年美国大学生数学建模竞赛MCM、ICM 试题 (38)2008 MCM A: Take a Bath (38)2008 MCM B: Creating Sudoku Puzzles (38)2008 ICM: Finding the Good in Health Care Systems (38)2009 年美国大学生数学建模竞赛MCM、ICM 试题 (40)2009 MCM A: Designing a Traffic Circle (40)2009 MCM B: Energy and the Cell Phone (40)2009 ICM: Creating Food Systems: Re-Balancing Human-Influenced Ecosystems41 2010年美国大学生数学建模竞赛 MCM、ICM 试题 (42)2010 MCM A: The Sweet Spot (42)2010 MCM B: Criminology (43)2010 ICM: The Great Pacific Ocean Garbage Patch (44)2011年美国大学生数学建模竞赛 MCM、ICM 试题 (45)2011 MCM A: Snowboard Course (45)2011 MCM B: Repeater Coordination (45)2011 ICM: Environmentally and Economically Sound (46)2012年美国大学生数学建模竞赛 MCM、ICM 试题 (48)2012 MCM A: The Leaves of a Tree (48)2012 MCM B: Camping along the Big Long River (50)2012 ICM: Modeling for Crime Busting (51)2013年美国大学生数学建模竞赛 MCM、ICM 试题 (59)2013 MCM A: The Ultimate Brownie Pan (59)2013 MCM B: Water, Water, Everywhere (61)2013 ICM: NetworkModeling of Earth's Health (62)2000 年美国大学生数学建模竞赛MCM、ICM 试题2000 MCM A: Air Traffic ControlTo improve safety and reduce air traffic controller workload, the Federal Aviation Agency (FAA) is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analysit at the FAA has posed the following problems.Requirement A: Given two airplanes flying in space, when should the air traffic controller consider the objects to be too close and to require intervention? Requirement B: And airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how do we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of aircraft simultaneously passing through that sector1.at any one instant?2.during any given interval of time?3.during a particular time of day?How does the number of potential conflicts arising during those periods affect complexity? Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this complexity? In addition to the guidelines for your report, write a summary (no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusions.2000 MCM B: Radio Channel AssignmentsWe seek to model the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grix (honeycomb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon.An interval of the frequency spectrum is to be alloted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1,2,3, … . Each transmitter wil be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided.Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assugn channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span.Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interference.Requirement A: There are several contrainsts on the frequency assignments. First, no two transmitters within distance 4s of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these contraints, what can we say about the span in Figure 1?Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions.Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance 2s differ by at least some given integer k, while those at distance at most 4s must still differ by at least one. What cna we say about the span and about efficient strategies for designing assignments, as a function of k?Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider?Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findings.2000 ICM: Elephants: When is Enough, Enough?“Ultimately, if a habitat is undesirably changed by elephants, then their removal should be considered -even by culling.”National Geographic (Earth Almanac) –December 1999 A large National Park in South Africa contains approximately 11,000 elephants. Management policy requires a healthy environment that can maintain a stable herf of 11,000 elephants. Each year park rangers count the elephant population. During the past 20 years whole herds have been removed to keep the population as close to 11,000 as possible. The process involved shooting (for the most part) and occasionally relocating approximately 600 to 800 elephants per year.Recently, there has been a public outcry against the shooting of these elephants. In addition, it is no longer feasible to relocate even a small population of elephants each year. A contraceptive dart, however, has been developed that can prevent a mature elephant cow from conceiving for a period of two years.Here is some information about eh elephants in the Park:∙There is very little emigration of immigration of elephants.∙The gender ratio is very close to 1:1 and control measures have endeavored to maintain parity.∙The gender ratio of newborn calves is also about 1:1. Twins are born about 1.35% of the time.∙Cows first conceive between the ages of 10 and 12 and produce, on average, a calf every 3.5 years until they reach an age of about 60.Gestation is approximately 22 months.∙The contraceptive dart causes an elephant cow to come into oestrus every month (but not conceiving). Elephants usually have courtship only once in 3.5 years, so the monthly cycle can cause additional stress.∙ A cow can be darted every year without additional detrimental effects. A mature elephant cow will not be able to conceive for 2 years after thelast darting.∙Between 70% and 80% of newborn calves survive to age 1 year.Thereafter, the survival rate is uniform across all ages and is very high(over 95%), until about age 60; it is a good assumption that elephantsdie before reading age 70.There is no hunting and negligible poaching in the Park.The park management has a rough data file of the approximate ages and gender of the elephants they have transported out of the region during the past 2 years. This data is available on website: icm2000data.xls. Unfortunately no data is available for the elephants that have been shot or remain in the Park.Your overall task is to develop and use models to investigate how the contraceptive dart might be used for population control. Specifically:Task 1: Develop and use a model to speculate about the likely survival rate for elephants aged 2 to 60. Also speculate about the current age structure of the elephant population.Task 2: Estimate how many cows would need to be darted each year to keep the population fixed at approximately 11,000 elephants. Show how the uncertainty in the data at your disposal affects your estimate. Comment on any changes in the age structure of the population and how this might affect tourists. (You may want to look ahead about 30-60 years.)Task 3: If it were feasible to relocate between 50 and 300 elephants per year, how would this reduce the number of elephants to be darted? Comment on the trade-off between darting and relocation.Task 4: Some opponents of darting argue that if there were a sudden loss of a large number of elephants (due to disease or uncontrolled poaching), even if darting stopped immediately, the ability of the population to grow again would be seriously impeded. Investigate and respond to this concer.Task 5: The management in the Park is skeptical about modeling. In particular, they argue that a lack of complete data makes a mockery of any attempt to use models to guide their decision. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the park management that responds to their concerns and provides advice. Also suggest ways to increase the park managers confidence in your model and your conclusions.Task 6: If your model works, other elephant parks in Africa would be interested in using it. Prepare a darting plan for parks of various sizes (300-25,000 elephants), with slightly different survival rates and transportation possibilities.2001 年美国大学生数学建模竞赛MCM、ICM 试题2001 MCM A: Choosing a Bicycle WheelCyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike.Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course.Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right.The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear.Task 1. Provide a table giving the wind speed at which the power required for a solid rear wheel is less than for a spoked rear wheel. The table should include the wind speeds for different road grades startingfrom zero percent to ten percent in one percent increments. (Roadgrade is defined to be the ratio of the total rise of a hill divided by thelength of the road. If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportionalto the road grade. A rider will lose about 8 kph for a five percent grade over 100 meters.∙Task 2. Provide an example of how the table could be used for a specific time trial course.∙Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision.2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...)Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare.The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned.A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of the coastal strip. (More accurate data, if sought, are widely available.)The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees (including some coming from farther north by other routes), so some traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only very slowly. Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed:1.Under what conditions does the plan for turning the two coastal-boundlanes of I-26 into two lanes of Columbia-bound traffic, essentiallyturning the entire I-26 into one-way traffic, significantly improveevacuation traffic flow?2.In 1999, the simultaneous evacuation of the state's entire coastal regionwas ordered. Would the evacuation traffic flow improve under analternative strategy that staggers the evacuation, perhapscounty-by-county over some time period consistent with the pattern of how hurricanes affect the coast?3.Several smaller highways besides I-26 extend inland from the coast.Under what conditions would it improve evacuation flow to turn around traffic on these?4.What effect would it have on evacuation flow to establish moretemporary shelters in Columbia, to reduce the traffic leaving Columbia?5.In 1999, many families leaving the coast brought along their boats,campers, and motor homes. Many drove all of their cars. Under whatconditions should there be restrictions on vehicle types or numbers ofvehicles brought in order to guarantee timely evacuation?6.It has been suggested that in 1999 some of the coastal residents ofGeorgia and Florida, who were fleeing the earlier predicted landfalls ofHurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation trafficflow? Clearly identify what measures of performance are used tocompare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.Clearly identify what measures of performance are used to compare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.2001 ICM: Our Waterways - An Uncertain FutureZebra mussels, Dreissena polymorpha, are small, fingernail-sized, freshwater mollusks unintentionally introduced to North America via ballast water from a transoceanic vessel. Since their introduction in the mid 1980s, they have spread through all of the Great Lakes and to an increasing number of inland waterways in the United States and Canada. Zebra mussels colonize on various surfaces,such as docks, boat hulls, commercial fishing nets, water intake pipes and valves, native mollusks and other zebra mussels. Their only known predators, some diving ducks, freshwater drum, carp, and sturgeon, are not numerous enough to have a significant effect on them. Zebra mussels have significantly impacted the Great Lakes ecosystem and economy. Many communities are trying to control or eliminate these aquatic pests. SOURCE: Great Lakes Sea Grant Network /.Researchers are attempting to identify the environmental variables related to the zebra mussel infestation in North American waterways. The relevant factors that may limit or prevent the spread of the zebra mussel are uncertain. You will have access to some reference data to include listings of several chemicals and substances in the water system that may affect the spread of the zebra mussel throughout waterways. Additionally, you can assume individual zebra mussels grow at a rate of 15 millimeters per year with a life span between 4 - 6 years. The typical mussel can filter 1 liter of water each day.Requirement A: Discuss environmental factors that could influence the spread of zebra mussels.Requirement B: Utilizing the chemical data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAChem1.xls, and the mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 1.xls model the population growth of zebra mussels in Lake A. Be sure to review the Information about the collection of the zebra mussel data. Requirement C: Utilizing additional data on Lake A from another scientist provided at :ap/undergraduate/contests/icm/imagesdata/LakeAChem2.xls and additional mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 2.xls corroborate the reasonableness of your model from Requirement B. As a result of this additional data, adjust your earlier model. Analyze the performance of your model. Discuss the sensitivity of your model. Requirement D: Utilizing the Chemical data from two lakes (Lake B and Lake C) in the United States provided atap/undergraduate/contests/icm/imagesdata/LakeB.xls and ap/undergraduate/contests/icm/imagesdata/LakeC.xls determine if these lakes are vulnerable to the spread of zebra mussels. Discuss your prediction.Requirement E: The community in the vicinity of Lake B (in requirement D) is considering specific policies for the de-icing of roadways near the lake duringthe winter season. Provide guidance to the local government officials regarding a policy on “de-icing agents.”In your guidance include predictions on the long-term impact of de-icing on the zebra mussel population. Requirement F: It has been recommended by a local community in the United States to introduce round goby fish. Zebra mussels are not often eaten by native fish species so they represent a dead end ecologically. However, round gobies greater than 100 mm feed almost exclusively on zebra mussels. Ironically, because of habitat destruction, the goby is endangered in its native habitat of the Black and Caspian Seas in Russia. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the local community leaders that responds to their recommendation to introduce the round goby. Also suggest ways to help reduce the growth of the mussel within and among waterways.Information about the collection of the zebra mussel dataThe developmental state of the Zebra mussel is categorized by three stages: veligers (larvae), settling juveniles, and adults. Veligers (microscopic zebra mussel larvae) are free-swimming, suspended in the water for one to three weeks, after which they begin searching for a hard surface to attach to and begin their adult life. Looking for zebra mussel veligers is difficult because they are not easily visible by the naked eye. Settled juvenile zebra mussels can be felt on smooth surfaces like boats and motors. An advanced zebra mussel infestation can cover a surface, even forming thick mats sometimes reaching very high densities. The density of juveniles was determined along the lake using three 15×15 cm settling plates. The top plate remained in the water for the entire sampling season (S - seasonal) to estimate seasonal accumulation. The middle and bottom plates are collected after specific periods (A –alternating ) of time denoted by “Lake Days”in the data files.The settling plates are placed under the microscope and all juveniles on the undersides of the plate are counted and densities are reported as juveniles/m^2.2002 年美国大学生数学建模竞赛MCM、ICM 试题2002 MCM A: Wind and WatersprayAn ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area. Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.2002 MCM B: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations. Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.Consider the overbooking issue in light of the current situation: Less flights by airlines from point A to point B Heightened security at and around airports Passengers' fear Loss of billions of dollars in revenue by airlines to dateBuild a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company's revenue is maximized. Insure that your model reflects the issues above, andconsider alternatives for handling “bumped”passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.2002 ICM: Scrub LizardsThe Florida scrub lizard is a small, gray or gray-brown lizard that lives throughout upland sandy areas in the Central and Atlantic coast regions of Florida. The Florida Committee on Rare and Endangered Plants classified the scrub lizard as endangered.You will find a fact sheet on the Florida Scrub Lizard at/undergraduate/contests/mcm/contests/2002/problem s/icm2002data/scrublizard.pdfThe long-term survival of the Florida scrub lizard is dependent upon preservation of the proper spatial configuration and size of scrub habitat patches.Task 1: Discuss factors that may contribute to the loss of appropriate habitat for scrub lizards in Florida. What recommendations would you make to the state of Florida to preserve these habitats and discuss obstacles to the implementation of your recommendations?Task 2: Utilize the data provided in Table 1 to estimate the value for Fa (the average fecundity of adult lizards); Sj (the survivorship of juvenile lizards- between birth and the first reproductive season); and Sa (the average adult survivorship).Table 1Summary data for a cohort of scrub lizards captured and followed for 4 consecutive years. Hatchling lizards (age 0) do not produce eggs during the summer they are born. Average clutch size for all other females is proportional to body size according to the function y = 0.21*(SVL)-7.5, where y is the clutch size and SVL is the snout-to-vent length in mm.Year Age Total NumberLivingNumber of LivingFemalesAvg. Female Size(mm)1 0 972 495 30.32 1 180 92 45.83 2 20 11 55.84 3 2 2 56.0Task 3: It has been conjectured that the parameters Fa , Sj , and Sa , are related to the size and amount of open sandy area of a scrub patch. Utilize the data provided in Table 2 to develop functions that estimate Fa, Sj , and Sa for different patches. In addition, develop a function that estimates C, the carrying capacity of scrub lizards for a given patch.Table 2Summary data for 8 scrub patches including vital rate data for scrub lizards. Annual female fecundity (Fa), juvenile survivorship (Sj), and adult survivorship (Sa) are presented for each patch along with patch size and the amount of open sandy habitat.Patch Patch Size (ha) Sandy Habitat (ha) Fa Sj Sa Density (lizards/ha)a 11.31 4.80 5.6 0.12 0.06 58b 35.54 11.31 6.6 0.16 0.10 60c 141.76 51.55 9.5 0.17 0.13 75d 14.65 7.55 4.8 0.15 0.09 55e 63.24 20.12 9.7 0.17 0.11 80f 132.35 54.14 9.9 0.18 0.14 82g 8.46 1.67 5.5 0.11 0.05 40h 278.26 84.32 11.0 0.19 0.15 115Task 4: There are many animal studies that indicate that food, space, shelter, or even reproductive partners may be limited within a habitat patch causing individuals to migrate between patches. There is no conclusive evidence on why scrub lizards migrate. However, about 10 percent of juvenile lizards do migrate between patches and this immigration can influence the size of the population within a patch. Adult lizards apparently do not migrate. Utilizing the data provided in the histogram below estimate the probability of lizards surviving the migration between any two patches i and patch j.Table 3HistogramMigration data for juvenile lizards marked, released, and recaptured up to 6 months later. Surveys for recapture were conducted up to 750m from release sites.Task 5: Develop a model to estimate the overall population size of scrub lizards for the landscape given in Table 3. Also, determine which patches are suitable for occupation by scrub lizards and which patches would not support a viable population.Patch size and amount of open sandy habitat for a landscape of 29 patches located on the Avon Park Air Force Range. See:/undergraduate/contests/icm/2002problem/map.jpg for a map of the landscape.Patch Identification Patch Size (ha) Sandy Habitat (ha)1 13.66 5.382 32.74 11.913 1.39 0.234 2.28 0.765 7.03 3.626 14.47 4.387 2.52 1.998 5.87 2.499 22.27 8.44。
2011AMC10美国数学竞赛B卷
2011 AMC10美国数学竞赛B 卷1. What is 246135135246++++-++++?(A) -1 (B) 536(C) 712(D)14760(E)4332. Josanna ’s test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise here test average at least 3 pints 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) 953. At a store, when a length is reported as x inches that means the length is at least x-0.5 inches and at most x+0.5 inches. Suppose the dimensions of a rectangular tile are reported as 2 inches by 3 inches. In square inches, what is the minimum area for the rectangle? (A) 3.75 (B)4.5 (C) 5 (D) 6 (E) 8.754. LeRoy 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 paid B dollars, where A<B. How many dollars must LeRoy give to Bernardo so that they share she costs equally? (A) 2A B + (B)2A B - (C)2B A - (D) B A - (E)A B+5. In multiplying two positive integers a and b, Ron reversed the digits of the two-digit number a. His erroneous product was 161. What is the correct value of the product of a and b?(A) 116 (B) 161 (C) 204 (D) 214 (E) 2246. On Halloween Casper ate 1/3 of his candies and then gave 2 candies to his brother. The next day he ate 1/3 of his remaining candies and then gave 4 candies to his sister. On the third day he ate his final 8 candies. How many candies did Casper have at the beginning?(A) 30 (B) 39 (C) 48 (D) 57 (E) 667. The sum of two angles of a triangle is 6/5 of a right angle, and one of these two angles is 30°larger than the other. What is the degree measure of the largest angle in the triangle?(A) 69 (B) 72 (C) 90 (D) 1024 (E) 1088. At a certain beach if it is at least 80℉and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?(A) The temperature was cooler than 80℉and it was not sunny.(B) The temperature was cooler than 80℉or it was not sunny.(C) If the temperature was at least 80℉, then it was sunny.(D) If the temperature was cooler than 80℉, then is was sunny. (E) If the temperature was cooler than 80℉, then it was not sunny.9. The area of △EBD is one third of the area of 3-4-5 △ABC. Segment DE is perpendicular to segment AB. What is BD?(A) 43(B)(C)94(D)3(E)5210. Consider the set of numbers {1, 10, 102, 103……1010}. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer? (A) 1 (B) 9(C) 10(D) 11 (E) 10111. There are 52 people in a room. What is the largest value of n such that the statement “At least n people in this room have birthdays falling in the same month ” is always true? (A) 2 (B) 3(C) 4(D) 5 (E) 1212. Keiko 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 a width of 6 meters, and it takes her 36 seconds 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) 3π(B)23π(C)π(D)43π (E)53π13. Two 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)2314. A rectangular parking lot has a diagonal of 25 meters and an area of 168 square meters. In meters, what is the perimeter of the parking lot? (A) 52(B) 58 (C) 62(D) 68(E) 7015. Let @ denote the “averaged with ” operation:@2a b a b +=. Which of the followingdistributive laws hold for all numbers x, y, and z? I. @()(@)@(@)x y z x y x z +=II. @()()@()x y z x y x z +=++ III.@(@)(@)@(@)x y z x y x z =(A) I only (B) II only (C) III only(D) I and III only (E) II and III only16. A 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)12(B)14(C)22-(D)4(E) 2-17. In the given circle, the diameter EB is parallel to DC, and AB is parallel to ED. The angles AEB and ABE are in the ratio 4:5. What is the degree measure of angle BCD?(A) 120 (B) 125 (C) 130(D) 135 (E) 14018. Rectangle ABCD has AB=6 and 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) 7519. What is the product of all the roots of the equation =(A) -64 (B) -24 (C) -9 (D) 24 (E) 57620. Rhombus ABCD has side length 2 and ∠B=120°. Region R consists of all points inside the rhombus that are closer to vertex B than any of the other three vertices. What is the area of R?(A)3(B)3(C)3(D) 13+(E) 221. Brian writes down four integers w>x>y>z whose sum is 44. The pairwise positiveBdifferences of these numbers are 1, 3, 4, 5, 6, and 9. What is the sum of the possible values for w? (A) 16 (B) 31 (C) 48 (D) 62 (E) 9322. A pyramid has a square base with sides of length land has lateral faces that are equilateral triangles. 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) 7(B)7- (C)27(D)9(E)923. What is the hundreds digit of 20112011?(A) 1 (B) 4 (C) 5 (D) 6 (E) 924. A lattice point in an xy-coordinate system in any point (x, y) where both x and y are integers. The graph of 2y m x =+ passes through no lattice point with0100x <≤for all m such that 12m a<<. What is the maximum possible value of a? (A) 51101(B)5099(C)51100(D)52101(E)132525. Let T 1 be a triangle with sides 2011, 2012, and 2013 for1n ≥, 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, respectively, then T n+1 is 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) 1509256。
2011年美国大学生数学建模竞赛优秀作品
AbstractThis paper presents one case study to illustrate how probability distribution and genetic algorithm and geographical analysis of serial crime conducted within a geographic information system can assist crime investigation.Techniques are illustrated for predicting the location of future crimes and for determining the possible residence of offenders based on the geographical pattern of the existing crimes and quantitative method,which is PSO.It is found that such methods are relatively easy to implement within GIS given appropriate data but rely on many assumptions regarding offenders’behaviour.While some success has been achieved in applying the techniques it is concluded that the methods are essentially theory-less and lack evaluation.Future research into the evaluation of such methods and in the geographic behaviour of serial offenders is required in order to apply such methods to investigations with confidence in their reliability.1.IntroductionThis series of armed robberies occurred in Phoenix,Arizona between13September and5December1999and included35robberies of fast food restaurants,hotels and retail businesses.The offenders were named the“Supersonics”by the Phoenix Police Department Robbery Detail as the first two robberies were of Sonic Drive-In restaurants.After the35th robbery,the offenders appear to have desisted from their activity and at present the case remains unsolved.The MO was for the offenders to target businesses where they could easily gain entry,pull on a ski mask or bandanna, confront employees with a weapon,order them to the ground,empty the cash from a safe or cash register into a bag and flee on foot most likely to a vehicle waiting nearby. While it appears that the offenders occasionally worked alone or in pairs,the MO, weapons and witness descriptions tend to suggest a group of at least three offenders. The objective of the analysis was to use the geographic distribution of the crimes to predict the location of the next crime in an area that was small enough to be suitable for the Robbery Detail to conduct stakeouts and surveillance.After working with a popular crime analysis manual(Gottleib,Arenberg and Singh,1994)it was found that the prescribed method produced target areas so large that they were not operationally useful.However,the approach was attractive as it required only basic information and relied on simple statistical analysis.To identify areas that were more useful for the Robbery Detail,it was decided to use a similar approach combined with other measurable aspects of the spatial distribution of the crimes.As this was a“live”case, new crimes and information were integrated into the analysis as it came to hand.2.AssumptionIn order to modify the model existed,we apply serial new assumptions to the principle so that our rectified model can be much more practical.Below are the assumptions:1.C riminals prefer something about the locations where previous crimes werecommitted committed..We supposed the criminals have a greater opportunity to ran away if they choose to crime in the site they are familiar with.In addition,the criminals probably choose previous kill sites where their target potential victims live and work.2.Offenders regard it safer to crime in their previous kill site as time went by.This is true that the site would be severely monitored by police when a short term crime happened and consequently the criminal would suffer a risk of being arrested in that site.And as mentioned above ,the police would reduce the frequency of examining the previous kill sites as time went by.3.Criminals are likely to choose the site that have optimal distance .This is a reasonable assumption since it is probably insecure to crime in the site that stays far away and that costs an amount of energy to escape and adds the opportunity to be arrested in such an unfamiliar terrain.And it is also impossible to crime in the site nearby since it increases the probability of being recognized or being trapped.As a result,we can measure a optimal distance in series perpetrations.4.Crimes are committed by individual.We assume that all the case in the model are committed by individuals instead of by organized members.In this way the criminal is subject to the assumptions mentioned above due to his insufficient preparation.5.Criminals Criminals''movements unconstrained.Because of the difficulty of finding real-world distance data,we invoke the “Manhattan assumption”:There are enough streets and sidewalks in a sufficiently grid-like pattern that movements along real-world movement routes is the same as “straight-line”movement in a space be discrete into city blocks.It is demonstrated that across several types of serial crime,the Euclidean and Manhattan distances are essentially interchangeable in predicting anchor points.3.The prediction of the next crime site3.1The measure of the optimal distanceDue to the fact that the mental optimal distance of the criminal is related to whether he is a careful person or not,it is impossible for him to make a fixed constant.Besides,the optimal distance will change in different moment.However,such distance should be reflected on the distances of the former crime sites.Presume that the coordinates of the n crime sites is respectively ),(11y x 、),(22y x 、……、),(n n y x ,and define the distance between the th i crime site and the th j one as j D ,i .The distance above we first consider it as Euclid distance,which is:22,)()(j i j i j i y y x x D −+−=With that,we are able to measure the distance between the th n crime site and the th 1-n one respectively.According to the assumption 2,the criminal believes that the earlier crime sites have became saferfor him to commit a crime again,so we can define his mental optimal distance,giving the sites the weights from little to much according to when the offenses happened in time sequence,as:∑−==11,n i ni i D w SD Satisfying 121......−<<<n w w w ，111=∑−=n i i w .Presuming the th i crime happens in i t ,whichis measured by week,we can have ∑−==11n i i kk t t w .SD can reflect the criminal's mental condition to some extent,so we can use it to predict the mental optimal distance of the criminal in the th n 1+case.While referring to the th n crime site,the criminal is able to use SD to estimate the optimal distance in the next time,and while referring to the rest crime sites,the optimal distances reduce as time goes back.Thus,the optimal security of the th i crime site can be measured as the following:n ni i SD t t SD *=3.2The measure of the probability distributionGiven the crime sites and location,we can estimate tentatively the probability density distribution of the future crimes,which equals to that we add some small normal distribution to every scene of crime to produce a probability distribution estimate function.The small normal distribution uses the SD mentioned above as the mean,which is:∑=−−=n i i i SD r n y x f 122)2)(exp(211),(σσπi r is defined as the Euclid distance between the site to the th i crime site,and the standard difference of the deviation of the criminal's mental optimal distance is defined as σ,which also reflects the uncertainty of the deviation of the criminal's mental optimal distance,involves the impacts of many factors and can not be measured quantitatively.The discussion of the standard difference is as following:3.3The quantization of the standard differenceThe standard difference is identified according to the following goal,which is,every prediction of the next crime site according to the crime sites where the crimes were committed before should have the highest rate of success.When having to satisfying such optimization objective,it isimpossible to make the direct analysis and exhaustivity.Instead,we have to use the optimized solutions searching algorithm,which is genetic algorithm.\Figure1:The Distribution of the Population of the Last GenerationAccording to the figure,the population of the last generation is mostly concentrated near80, which is used as the standard distance and substituted to the*formula.With the*formula,we are able to predict the probability density of Whether the zones will be the next crime site.Case analysis:5crime site according to the4ones happened before Figure2:The prediction of theth6crime site according to the5ones happened before Figure3:The prediction of theth6crime site according to the5ones happened before Figure4:The prediction of thethAccording to the predictions happened before,the predictions of the outputs based on the models are accurate relatively,and they are able to be the references of the criminal investigations to some extent.However,when is frequency of such crime increases,the predictions of the outputs23crime site according deviated the actual sites more and more,such as the prediction of thethto the22ones happened before,which is:23crime site according to the22ones happened before Figure5:the prediction of thethConclusion according to analysis:It may not be able to predict the next crime site accurately if we use Euclid distance to measure the probability directly.So,we should analyze according to the actual related conditions.For example,we can consider the traffic commutes comprehensively based on the conveniences of the escapes,such as the facilities of the express ways network and the tunnels.According to the hidden security of the commitments,we should consider the population of the area and the distance from the police department.Thus,we should give more weights to the commute convenience,hidden security and less population.In addition,when the commitments increases,the accuracy of the model may decrease,resulted from the fact that when the criminal has more experience,he will choose the next crime sites more randomly.4.Problems and further improvementsWith23crimes in the series the predictions tended to provide large areas that included the target crime but were too large to be useful given the limited resources the police had at their disposal.At this stage,a more detailed look was taken at the directionality and distances between crimes.No significant trends could be found in the sequential distance between crimes so an attempt was made to better quantify the relationship between crimes in terms of directionality.The methodology began by calculating the geographic center of the existing crimes. The geographic center is a derived point that identifies the position at which the distance to each crime is minimized.For applications of the geographic center to crime analysis.Once constructed,the angle of each crime from the north point of the geographic center was calculated.From this it was possible to calculate the change indirection for the sequential crimes.It was found that the offenders were tending to pattern their crimes by switching direction away from the last crime.It appears that the offenders were trying to create a random pattern to avoid detection but unwittingly created a uniform pattern based upon their choice of locations.This relationship was quantified and a simple linear regression used to predict what the next direction would be.The analysis was once again applied to the data.While the area identified was reduced from previous versions and prioritized into sub-segments,the problem remained that the areas predicted were still too large to be used as more than a general guide to resource deployment.A major improvement to the methodology was to include individual targets.By this stage of the series,hotels and auto parts retailers had become the targets of choice.A geo-coded data set became available that allowed hotels and retail outlets to be plotted and compared to the predicted target areas.Ideally those businesses falling within the target areas could be prioritized as more likely targets.However,in some cases the distribution of the likely businesses appeared to contradict the area predicted.For example,few target hotels appeared in the target zone identified by the geographic analysis.In this case,more reliance was placed upon the location of individual targets. From this analysis it was possible to identify a prioritized list of individual commercial targets,which was of more use operationally.Maps were also provided to give an indication of target areas.Figure6demonstrates a map created using this methodology.It is apparent from the above discussion that the target areas identified were often too large to be used as more than a general guide by the Robbery Detail.However,by including the individual targets,it was possible to restrict the possible target areas to smaller,more useful areas,and a few prioritized targets.However,such an approach has the danger of being overly restrictive and it is not the purpose of the analysis to restrict police operations but to suggest priorities.This problem was somewhat dealt with by involving investigators in the analysis and presenting the results in an objective manner,such that investigators could make their own judgments about the results.To be more confident in using this kind of analysis a stronger theoretical background to the methods is required.What has been applied here is to simply exploit the spatial relationships in the information available without considering what the connection is to the actual behaviour of the offenders.For example,what is the reason behind a particular trend observed in the distance between crimes?Why would such a trend be expected between crimes that occur on different days and possibly involve different individuals?While some consideration was given to identifying the reason behind the pattern of directionality and while it seems reasonable to expect offender’s to look for freeway access,such reasoning has tended to follow the analysis rather than substantiate it.Without a theoretical background the analysis rests only on untested statistical relationships that do not provide an answer to the basic question:why this pattern?So next we will apply a quantitative method,which is PSO,based on a theoretical background,to locate the residence of the criminal's residence.5.The prediction of the residenceParticle Swarm Optimization is a evolutionary computation,invented by Dr.Eberhart and Dr.Kennedy.It is a tool of optimization based on iteration,resulted from the research on the behaviors of the bird predation.Initiating a series of random number,the PSO is able to catch the optimization with iteration.Like PSO,the resolution of our residence search problem is the criminal,whose serial crime sites have been abstracted into 23particles without volume and weight and extended to the 2-D space.Like bird,the criminal is presumed to go directly home when he committed a crime.So,there are 23criminals who commit the crimes in the 23sites mention before and then they will go home directly.The criminals are defined as a vector,so are their speed.All criminals have a fittness decided by the optimized functions,and every of them has a according speed which can decide their direction and distance.All the criminals know the best position (pbest,defined as the residence known by the individual),which has been discovered so far,and where they are now.Besides,every criminals also know the best position which has been found by the group (gbest,defined as the residence known by the group).Such search can be regarded as the experience of other criminals.The criminals are able to locate the residence by the experience of itself and the whole criminals.PSO computation initiates the 23criminals and then the offenders will pursue the optimized one to search in the space.In other words,they find the optimized solutions by iteration.Presume that in the 2-D space the location and speed of the ith crime site is relatively ),(2,1,i i i x x X =and ),(2,1,i i i v v V =.In every iteration,the criminals will pursue the two best positions to update themselves.The two best positions are relatively the individual peak (pbest),),(2,1,i i i p p P =,which is found by the criminal himself,and the group optimized solution (gbest),g P ,which has been found to be the optimized solution by the whole group so far.When the criminals found the two optimized solutions,they will update their speed and new position based on the following formulas.2,1),1()()1()]([)]([)()1(,,,,,22,,11,,=++=+−+−+=+j t v t x t x t x p r c t x p r c t wv t V j i j i j i j i j g j i j i j i j i In the above,the w is inertial weighted factor,21c andc are positive learning factors,21r andr are random number which are distributed uniformly between 0and 1.The learning factor can make the criminals have self-conclude ability and ability of learning from others.Here we make both of them be 2,as what they always are in PSO.The inertial weighted factor w decides the extent of the inheritance of the current speed of the crime sites.The appropriate choice can make them have balanced searching and exploring ability.For balancing the global searching ability and the local improving ability of the criminal in the PSO algorithm,here we adopt one of the self-adapted methods,which is Non-linear Dynamic Inertial Weight Coefficient to choose the inertial weight.The expression is as following:⎪⎩⎪⎨⎧=≤−−−−>avg avg avg f f f f f f w w w f f w w ,))*((,minmin min max min max In the above,the max w and min w are defined respectively as the maximum and minimum of w,f means the current functional value of the criminal,and the avg f and min f respectively means the average value and minimum value of all the current criminals.In addition,the inertial weight will change automatically according to the objective value,which gives the name self-adapted method.When the final values,which are estimations of the criminal's residence,become consistent,it will make the inertial weight increase.When they become sparser,it will make the inertial weight decrease.In the meantime,referring to the criminals whose final values are worse than the average value,its according inertial weighted factor will become smaller,which protect the crime site.Oppositely,when referring to the criminals whose final values are better than the average value,its according inertial weighted factor will become bigger,which makes the criminal nearer to the searching zone.So now,with the PSO of Non-linear Dynamic Inertial Weight Coefficient,we can calculate the minimum value of22,)()(j j j i y y x x R −+−=,j=1,2,3 (23)In the above,j ,i R is the residence of the criminal.Thus,we have the output (x,y)as(2.368260870656715,3.031739124610613).We can see the residence in the figure 7.Figure7:The residence in the map6.ConclusionThis paper has presented one case study to illustrate how probability distribution and geographical analysis of serial crime conducted can assist crime investigation. Unfortunately,in the Supersonic armed robbery investigation the areas identified were too large to have been of much use to investigators.Further,because of the number of assumptions applied the method does not inspire enough confidence to dedicate resources to comparing its results to the enormous amount of suspect data collected on the case.While the target areas predicted tended to be large,the mapping of individual commercial targets appears to offer a significant improvement to the method.However,as they stand,these methods lack a theoretical basis that would allow the results to be judged and applied in investigations.Limitations such as these can be offset to some degree by the involvement of investigators in the analysis.In the end,we used a quantitative method to locate the residence of the criminal to make the identified areas smaller.So,due to the advantages and drawbacks of the above methods,we suggest that we should use different methods to help us fight again the crimes comprehensively.。
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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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