美国大学生数学竞赛2011题目中文版

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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。

2011年-AMC10数学竞赛A卷-附中文翻译和答案

2011年A M C1 0美国数学竞赛A 卷1. A cell phone plan costs $20 each month, plus 5¢per text message sent, plus 10¢for each minute used over 30 hours. In January Michelle sent 100 text messages and talked for 30.5 hours. How much did she have to pay?(A) $24.00 (B) $24.50 (C) $25.50 (D) $28.00 (E) $30.002. A small bottle of shampoo can hold 35 milliliters of shampoo, Whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?(A) 11 (B) 12 (C) 13 (D) 14 (E) 153. Suppose [a b] denotes the average of a and b, and {a b c} denotes the average of a, b, and c. What is {{1 1 0} [0 1] 0}?(A) 29(B)518(C)13(D) 718(E) 234. Let X and Y be the following sums of arithmetic sequences: X= 10 + 12 + 14 + … + 100.Y= 12 + 14 + 16 + … + 102.What is the value of Y X?(A) 92 (B) 98 (C) 100 (D) 102(E) 1125. At an elementary school , the students in third grade, fourth grade, and fifth grade run an average of 12, 15 , and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?(A) 12 (B) 373 (C) 887 (D) 13 (E) 146. Set A has 20 elements, and set B has 15 elements. What is the smallest possible number of elements in A ∪B, the union of A and B?(A) 5 (B) 15 (C) 20 (D) 35 (E) 3007. Which of the following equations does NOT have a solution?(A) 2(7)0x += (B) -350x += (C) 20=(D) 80= (E) -340x -=8. Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons , and 35% were ducks. What percent of the birds that were not swans were geese?(A) 20 (B) 30 (C) 40 (D) 50 (E) 609. A rectangular region is bounded by the graphs of the equations y=a, y=-b, x=-c, and x=d, where a, b, c, and d are all positive numbers. Which of the following represents the area of this region?(A) ac + ad + bc + bd(B) ac – ad + bc – bd (C) ac + ad – bc – bd(D) –ac –ad + bc + bd (E) ac – ad – bc + bd10. A majority of the 20 students in Ms. Deameanor’s class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. 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) 7711. Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7·EB. What is the ratio of the area of EFGH to the area of ABCD?(A) 4964 (B) 2532 (C) 78 (D) 8 (E)12. The players on a basketball team made some three-point shots, some two-point shots, some one-point free t hrows. 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 t wo-point shots. The team’s total score was 61 points. Howmany free throws did they m ake?(A) 13 (B) 14 (C) 15 (D) 16 (E) 1713. How many even integers are there between 200 and 700 whose digits are all different and come from the set {1, 2, 5, 7, 8, 9}?(A) 12 (B)20 (C)72 (D) 120 (E) 20014. A pair of standard 6-sided fai r 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 th e numerical value of the circle’s circumference?(A) 136(B)112(C)16(D) 14(E) 5181 5. Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on I ts battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 m iles per gallon. How long was the trip in miles?(A) 140 (B) 240 (C) 440 (D) 640 (E) 84016. Wh ich of the following in equal to(A) (B) (C) 2 (D) (E) 617. In the eight-term sequence A, B, C, D, E, F, G, H, the value of C is 5 and the sum o f any three consecutive terms is30. What is A + H?(A) 17 (B) 18 (C) 25 (D) 26 (E) 431 8. Circles A, B, and C each have radius 1. Circles A and B share one point of ta ngency. Circle C has a point of tangency with the midpoint of AB. What is the area inside Circle C but outside Circle A and Circle B? (A) 32π- (B) 2π (C) 2 (D) 34π (E) 12π+1 9. In 1991 the population of a town was a perfect square. Ten years later, after an in crease of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town’s population during this twenty-year period?(A) 42 (B) 47 (C) 52 (D) 57 (E) 622 0. Two points on the circumference of a circle of radius r are selected independently an d at random. From each point a chord of length r is drawn in a clockwise direction. Wh at is the probability that the two chords intersect?(A) 16(B) 15(C) 14(D) 13(E) 122 1. Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. T he weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?(A) 711(B) 913(C) 1115(D) 1519(E) 15162 2. Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 co lors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?(A) 2500 (B) 2880 (C) 3120 (D) 3250 (E) 37502 3. Seven students count from 1 to 1000 as follows:·Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is Alice says 1, 3, 4, 6, 7, 9, …, 997, 999, 1000.·Barbara says all of the numbers that Alice doesn’t say, except she also skips the middle number in each consecutive grope of three numbers.·Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middlenumber in each consecutive group of three numbers.· Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.· Finally, George says the only number that no one else says.Wh at number does George say?(A) 37 (B) 242 (C) 365 (D) 728 (E) 9982 4. Two distinct regular tetrahedra have all their vertices among the vertices of the sa me unit cube. What is the volume of the region formed by the intersection of the tetrahedra?(A) 112 (B) 12 (C) (D)16 (E) 62 5. Let R be a square region and 4n an integer. A point X in the interior of R is ca lled n-ray partitional 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) 25002011AMC10美国数学竞赛A卷1. 某通讯公司手机每个月基本费为20美元, 每传送一则简讯收5美分（一美元=100 美分）。

2011AMC10美国数学竞赛A卷附中文翻译和答案

2011A M C10美国数学竞赛A卷附中文翻译和答案2011AMC10美国数学竞赛A卷1. A cell phone plan costs $20 each month, plus 5¢ per text message sent, plus 10¢ for each minute used over 30 hours. In January Michelle sent 100 text messages and talked for 30.5 hours. How much did she have to pay?(A) $24.00 (B) $24.50 (C) $25.50 (D) $28.00 (E) $30.002. A small bottle of shampoo can hold 35 milliliters of shampoo, Whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?(A) 11 (B) 12 (C) 13 (D) 14 (E) 153. Suppose [a b] denotes the average of a and b, and {a b c} denotes the average of a, b, and c. What is {{1 1 0} [0 1] 0}?(A) 29(B)518(C)13(D) 718(E) 234. Let X and Y be the following sums of arithmetic sequences:X= 10 + 12 + 14 + …+ 100.Y= 12 + 14 + 16 + …+ 102.What is the value of Y X?(A) 92 (B) 98 (C) 100 (D) 102 (E) 1125. At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively. There are twice asmany third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?(A) 12(B) 373 (C) 887 (D) 13 (E) 146. Set A has 20 elements, and set B has 15 elements. What is the smallest possible number of elements in A ∪B, the union of A and B?(A) 5(B) 15 (C) 20 (D) 35 (E) 3007. Which of the following equations does NOT have a solution?(A)2(7)0x +=(B) -350x += 20=80=(E) -340x -=8. Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?(A) 20(B) 30 (C) 40 (D) 50 (E) 609. A rectangular region is bounded by the graphs of the equations y=a, y=-b, x=-c, and x=d, where a, b, c, and d are all positive numbers. Which of the following represents the area of this region?(A) ac + ad + bc + bd(B) ac – ad + bc – bd (C) ac + ad – bc – bd (D) –ac –ad + bc + bd(E) ac – ad – bc + bd10. A majority of the 20 students in Ms. Deameanor’s class bought pencils at theschool bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. 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) 7711. Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7·EB. What is the ratio of the area of EFGH to the area of ABCD?(A)4964 (B) 2532 (C) 78 (D) 8 (E) 412. The players on a basketball team made some three-point shots, some two-pointshots, 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 was 61 points. Howmany free throws did they make?(A) 13 (B) 14 (C) 15 (D) 16 (E) 1713. How many even integers are there between 200 and 700 whose digits are all different and come from the set {1, 2, 5, 7, 8, 9}?(A) 12 (B)20 (C)72 (D) 120 (E) 20014. A pair of standard 6-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) 51815. Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged55 miles per gallon. How long was the trip in miles?(A) 140 (B) 240 (C) 440 (D) 640 (E) 84016. Which of the following in equal to(A)(B) (D) (E) 617. In the eight-term sequence A, B, C, D, E, F, G, H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A + H?(A) 17(B) 18 (C) 25 (D) 26 (E) 4318. Circles A, B, and C each have radius 1. Circles A and B share one point oftangency. Circle C has a point of tangency with the midpoint of AB. What is the area inside Circle C but outside Circle A and Circle B? (A) 32π- (B) 2π(C) 2 (D) 34π (E) 12π+19. In 1991 the population of a town was a perfect square. Ten years later, after anincrease of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the pe rcent growth of the town’s populationduring this twenty-year period?(A) 42(B) 47 (C) 52 (D) 57 (E) 6220. Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect? (A) 16 (B) 15 (C) 14 (D) 13 (E) 1221. Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?(A) 711(B) 913(C) 1115(D) 1519(E) 151622. Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?(A) 2500 (B) 2880 (C) 3120 (D) 3250 (E) 375023. Seven students count from 1 to 1000 as follows:·Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is Alice says 1, 3, 4, 6, 7, 9, …, 997, 999, 1000.·Barbara says all of the numbers that Alice doesn’t say, except she also skips the middle number in each consecutive grope of three numbers.·Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers. ·Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.·Finally, George says the only number that no one else says.What number does George say?(A) 37(B) 242 (C) 365 (D) 728 (E) 99824. Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?(A)112 (B) 12 (C) 12 (D) 16 (E) 625. Let R be a square region and 4n an integer. A point X in the interior of R is called n-ray partitional if there are n rays emanating from X that divide R into Ntriangles of equal area. How many points are 100-ray partitional but not 60-raypartitional?(A) 1500(B) 1560 (C) 2320 (D) 2480 (E) 25002011AMC10美国数学竞赛A 卷1. 某通讯公司手机每个月基本费为20美元, 每传送一则简讯收 5美分（一美元=100 美分）。

1998-2014年历年美赛题目中文版!!!

2014年美赛题目翻译问题A：除非超车否则靠右行驶的交通规则在一些汽车靠右行驶的国家（比如美国，中国等等），多车道的高速公路常常遵循以下原则：司机必须在最右侧驾驶，除非他们正在超车，超车时必须先移到左侧车道在超车后再返回。

建立数学模型来分析这条规则在低负荷和高负荷状态下的交通路况的表现。

你不妨考察一下流量和安全的权衡问题,车速过高过低的限制，或者这个问题陈述中可能出现的其他因素。

这条规则在提升车流量的方面是否有效？如果不是，提出能够提升车流量、安全系数或其他因素的替代品（包括完全没有这种规律）并加以分析。

在一些国家，汽车靠左形式是常态，探讨你的解决方案是否稍作修改即可适用，或者需要一些额外的需要。

最后，以上规则依赖于人的判断，如果相同规则的交通运输完全在智能系统的控制下，无论是部分网络还是嵌入使用的车辆的设计，在何种程度上会修改你前面的结果？问题B：大学传奇教练体育画报是一个为运动爱好者服务的杂志，正在寻找在整个上个世纪的“史上最好的大学教练”。

建立数学模型选择大学中在一下体育项目中最好的教练：曲棍球或场地曲棍球，足球，棒球或垒球，篮球，足球。

时间轴在你的分析中是否会有影响？比如1913年的教练和2013年的教练评价是否会有所不同？清晰的对你的指标进行评估，讨论一下你的模型应用在跨越性别和所有可能对的体育项目中的效果。

展示你的模型中的在三种不同体育项目中的前五名教练。

除了传统的MCM格式，准备一个1到2页的文章给体育画报，解释你的结果和包括一个体育迷都明白的数学模型的非技术性解释。

2013年美赛题目翻译A ：当用方形的平底锅烤饼时，热量会集中在四角，食物就在四角（甚至还有边缘）烤焦了。

在一个圆形的平底锅热量会均匀分布在整个外缘，食物就不会被边缘烤焦。

但是，因为大多数烤箱是矩形的，使用圆形的平底锅不那么有效率。

建立一个模型来表现热量在不同形状的平底锅的外缘的分布——包括从矩形到圆形以及中间的形状。

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。

美国大学生数学建模竞赛试题AB题中文

A 题热水澡一个人进入浴缸洗澡放松。

浴缸的热水由一个水龙头放出。

然而浴缸不是一个可以水疗泡澡的缸，没有辅助加热系统和循环喷头，仅仅就是一个简单的盛水容器。

过一会，水温就会显著下降。

因此必须从热水龙头里面反复放水以加热水温。

浴缸的设计就是当水达到浴缸的最大容量，多余的水就会通过一个溢流口流出。

做一个有关浴缸水温的模型，从时间和地点两个方面来确定在浴缸中泡澡的人能采用的最佳策略，从而泡澡过程中能保持水温并在不浪费太多水的情况下使水温尽量接近最初的水温。

用你的模型来确定你的策略多大程度上依赖于浴缸的形状和容量，浴缸中的人的体型/体重/体温，以及这个人在浴缸中做出的动作。

如果这个人在最开始放水的时候加入了泡泡浴添加剂，这将会对你的模型结果有什么影响？要求提交一页MCM的总结，此外你的报告必须包括一页给浴缸用户看的非技术性的解释，其中描述了你的策略并解释了在泡澡过程中为什么保持平均的水温会非常困难。

B题太空垃圾地球轨道周围的小碎片的数量受到越来越多的关注。

据估计，目前大约有超过50万片太空碎片被视为是宇宙飞行器的潜在威胁并受到跟踪，这些碎片也叫轨道碎片。

2009年2月10号俄罗斯卫星科斯莫斯-2251与美国卫星iridium-33相撞的时候，这个问题在新闻媒体上就愈发受到广泛讨论。

已经提出了一些方法来清除这些碎片。

这些方法包括小型太空水流喷射器和高能量激光来瞄准具体的碎片，还有大型卫星来清扫碎片等等。

这些碎片数量和大小不一，有油漆脱离的碎片，也有废弃的卫星。

碎片高速转动使得定位清除变得困难。

建一个随时间变化的模型来确定一个最佳选择或组合的选择提供给一家私人公司让它以此为商业机遇来解决太空碎片问题。

你的模型应该包括对成本、风险、收益的定量和/或定性分析以及其他重要因素的分析。

你的模型应该既能够评估单个的选择也能够评估组合的选择，且能够探讨一些重要的”what if ”情景。

用你的模型来确定是否存在这样的机会，在经济上很有吸引力；或是根本不可能有这样的机会。

2011AMC10美国数学竞赛A卷中文翻译及答案

2011AMC10美国数学竞赛A卷1. 某通讯公司手机每个月基本费为20美元, 每传送一则简讯收 5美分（一美元=100 美分）。

若通话超过30小时，超过的时间每分钟加收10美分。

已知小美一月份共传送了100条简讯及通话30.5小时，则她需要付多少美元？(A) $24.00 (B) $24.50 (C) $25.50 (D) $28.00 (E) $30.002.小瓶装有35毫升的洗发液，大瓶可装500毫升的洗发液。

小华至少要买多少瓶小瓶的洗发液才能装满一个大瓶的洗发液?(A) 11 (B) 12 (C) 13 (D) 14 (E) 153. 若以 [a b]表示 a ， b两数的平均数, 以 {a b c} 表示a, b, c三数的平均数，则{{1 1 0} [0 1] 0}之值为何?(A) 29(B)518(C)13(D) 718(E) 234. 设 X 和 Y 为下列等差级数之和:X= 10 + 12 + 14 + …+ 100.Y= 12 + 14 + 16 + …+ 102.则Y X之值为何?(A) 92 (B) 98 (C) 100 (D) 102 (E) 1125. 在某小学三年级，四年级及五年级的学生，每天分别平均跑12, 15, 及10 分钟, 已知三年级的学生人数是四年级人数的两倍，四年级的学生人数是五年级学生人数的两倍。

试问所有这些学生每天平均跑几分钟？(A) 12 (B) 373 (C) 887 (D) 13 (E) 146. 已知集合A 中有20个元素, 集合B 中有 15 个元素. A ∪B 是集合A 和集合B 的联集,它是由集合A 与集合B 中所有元素所形成的集合，则集合A ∪B 中至少有多少个元素?(A) 5(B) 15 (C) 20 (D) 35 (E) 3007.下列哪个方程式没有解?(A) 2(7)0x += (B) -350x += 20=80= (E) -340x -=8.去年夏季保护区里有 30%是鹅，25%是鸳鸯, 10%是苍鹰, 35% 是鸭子. 试问不是鸳鸯的鸟类中鹅占多少百分比？(A) 20(B) 30 (C) 40 (D) 50 (E) 609. 某个矩形是由y=a, y=-b, x=-c, 与x=d,的圆形所围成的，其中a, b, c, ， d 均为正数。

2004、2010、2011、2012美国数学建模数学竞赛试题及翻译、分析

2004年A题题目是研究人的指纹相同导致确认身份时产生错误的可能性和因为DNA相同导致产生错误的可能性。

这道题与生物有关。

2004年，美国提出了“科技展望——生物盾国家纳米技术”，针对微分子进行研究，与人体相关的DNA也就成为焦点。

指纹是区分人的重要参数，它由DNA决定，由于现在自然环境改变，人体的DNA也有可能发生变异，这样引起相同指纹出现也是有可能的。

研究这种变异的可能性有利于刑事案件的侦破，提高政府的防御能力等。

本题目可以采用机理分析法来建模。

可以采用计算机模拟人体染色体中的基因排列进而检验、统计排列结果与指纹形成情况B题题目：现在的快通系统在收费站、娱乐公园和其他的地方，正在越来越频繁的使用，来减少人们排队等候的时间，现在我们考虑为一个娱乐公园所设计的快通系统，在一次测试中，这个公园在几个游客比较多的景点旁都设置的快通系统，这个系统的设计创意是对于那些比较热门的景点，可以到旁边的一个机器，将其门票插入后出来一张纸条，上面写着具体的你可以回来时间段，比如说你把你的门票在1：15插到机子里，系统就会告诉你，你可以在3:30-4:30回来，这个时候队伍就比较短，你可以凭你的纸条加入这个队伍，很快就可以进入景点，为了防止游客同时在几个景点使用这个系统。

系统的机器只允许你一次在一个景点排队等候。

现在你是几个被公园雇佣的互相竞争的一个，你的职责是改善快通系统的运行。

很多游客都在抱怨测试期间系统的异常现象，比如说有一次系统提供的回到景点的时间是4小时以后，但是才过一会儿，在相同景点系统提供的时间只有1小时。

在另外一些时候根据快通系统组织起来的游客的等候队伍，就和普通的队伍一样长一样慢。

现在的问题是要提出并且测试一个模型，这个模型能让快通系统的等候纸条的发放能增加人们在公园的乐趣的目的。

问题的一部分就是首先决定衡量不同模型的标准，在你提交的报告里还要附带一份技术性的总结，以便公园的领导在不同的顾问所提出的模型中选择。

密苏里州大学生数学竞赛2011试题答案

1 3 5 · · 2 4 6
·
x7 + ··· . 7
Therefore, for each 0 < x ≤ 1, there exists a θ1 and θ2 between 0 and x such that
5 x3 θ1 + 3! 5! 1 x3 arcsin x = x + · + 2 3
2011 Missouri Collegiate Mathematics Competition Session I Solutions Problem 1. Consider the parabola x2 = 4 p y , with vertex at the origin, focus at F (0, p), and directrix y = −p. Let P (x, y ) be a point on the parabola and let R(x, −p) be the intersection of the vertical line through P with the directrix. Determine P so that the triangle P F R is equilateral. Solution: By the deﬁnition of a parabola, dist(F, P ) = dist(P, R) = dist(F, R) = 4p2 + x2 , so we need x2 + 4 p 2 = 4p which is equivalent to √ making x = 2 3p and y = 3p. 4p2 + x2 = 4p 4p2 + x2 x2 + 4 p 2 . 4p

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。

AMC8(美国数学竞赛)历年真题、答案及中英文解析

AMC8（美国数学竞赛）历年真题、答案及中英文解析艾蕾特教育的AMC8 美国数学竞赛考试历年真题、答案及中英文解析：AMC8-2020年：真题 --- 答案---解析（英文解析+中文解析）AMC8 - 2019年：真题----答案----解析（英文解析+中文解析）AMC8 - 2018年：真题----答案----解析（英文解析+中文解析）AMC8 - 2017年：真题----答案----解析（英文解析+中文解析）AMC8 - 2016年：真题----答案----解析（英文解析+中文解析）AMC8 - 2015年：真题----答案----解析（英文解析+中文解析）AMC8 - 2014年：真题----答案----解析（英文解析+中文解析）AMC8 - 2013年：真题----答案----解析（英文解析+中文解析）AMC8 - 2012年：真题----答案----解析（英文解析+中文解析）析）AMC8 - 2010年：真题----答案----解析（英文解析+中文解析）AMC8 - 2009年：真题----答案----解析（英文解析+中文解析）AMC8 - 2008年：真题----答案----解析（英文解析+中文解析）AMC8 - 2007年：真题----答案----解析（英文解析+中文解析）AMC8 - 2006年：真题----答案----解析（英文解析+中文解析）AMC8 - 2005年：真题----答案----解析（英文解析+中文解析）AMC8 - 2004年：真题----答案----解析（英文解析+中文解析）AMC8 - 2003年：真题----答案----解析（英文解析+中文解析）AMC8 - 2002年：真题----答案----解析（英文解析+中文解析）AMC8 - 2001年：真题----答案----解析（英文解析+中文解析）AMC8 - 2000年：真题----答案----解析（英文解析+中文解析）析）AMC8 - 1998年：真题----答案----解析（英文解析+中文解析）AMC8 - 1997年：真题----答案----解析（英文解析+中文解析）AMC8 - 1996年：真题----答案----解析（英文解析+中文解析）AMC8 - 1995年：真题----答案----解析（英文解析+中文解析）AMC8 - 1994年：真题----答案----解析（英文解析+中文解析）AMC8 - 1993年：真题----答案----解析（英文解析+中文解析）AMC8 - 1992年：真题----答案----解析（英文解析+中文解析）AMC8 - 1991年：真题----答案----解析（英文解析+中文解析）AMC8 - 1990年：真题----答案----解析（英文解析+中文解析）AMC8 - 1989年：真题----答案----解析（英文解析+中文解析）AMC8 - 1988年：真题----答案----解析（英文解析+中文解析）析）AMC8 - 1986年：真题----答案----解析（英文解析+中文解析）AMC8 - 1985年：真题----答案----解析（英文解析+中文解析）◆AMC介绍◆AMC（American Mathematics Competitions) 由美国数学协会（MAA）组织的数学竞赛，分为 AMC8 、 AMC10、 AMC12 。

2011AMC10美国数学竞赛A卷附中文翻译和答案

2011AMC10美国数学竞赛A卷1. A cell phone plan costs $20 each month, plus 5¢ per text message sent, plus 10¢ for each minute used over 30 hours. In January Michelle sent 100 text messages and talked for 30.5 hours. How much did she have to pay?(A) $24.00 (B) $24.50 (C) $25.50 (D) $28.00 (E) $30.002. A small bottle of shampoo can hold 35 milliliters of shampoo, Whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?(A) 11 (B) 12 (C) 13 (D) 14 (E) 153. Suppose [a b] denotes the average of a and b, and {a b c} denotes the average of a, b, and c. What is {{1 1 0} [0 1] 0}?(A) 29(B)518(C)13(D) 718(E) 234. Let X and Y be the following sums of arithmetic sequences: X= 10 + 12 + 14 + …+ 100.Y= 12 + 14 + 16 + …+ 102.What is the value of Y X?(A) 92 (B) 98 (C) 100 (D) 102 (E) 1125. At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?(A) 12 (B) 373 (C) 887 (D) 13 (E) 146. Set A has 20 elements, and set B has 15 elements. What is the smallest possible number of elements in A ∪B, the union of A and B?(A) 5(B) 15 (C) 20 (D) 35 (E) 3007. Which of the following equations does NOT have a solution?(A)2(7)0x +=(B) -350x += (C) 20= (D)80= (E) -340x -=8. Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?(A) 20(B) 30 (C) 40 (D) 50 (E) 609. A rectangular region is bounded by the graphs of the equations y=a, y=-b, x=-c, and x=d, where a, b, c, and d are all positive numbers. Which of the following represents the area of this region?(A) ac + ad + bc + bd (B) ac – ad + bc – bd (C) ac + ad – bc – bd(D) –ac –ad + bc + bd (E) ac – ad – bc + bd10. A majority of the 20 students in Ms. Deameanor’s class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. 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) 7711. Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7·EB. What is the ratio of the area of EFGH to the area of ABCD?(A)4964 (B) 2532 (C) 78 (D) (E)12. The players on a basketball team made some three-point shots, some two-point shots, 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 was 61 points. How many free throws did they make?(A) 13(B) 14 (C) 15 (D) 16 (E) 1713. How many even integers are there between 200 and 700 whose digits are alldifferent and come from the set {1, 2, 5, 7, 8, 9}?(A) 12(B) 20 (C) 72 (D) 120 (E) 20014. A pair of standard 6-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) 51815. Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?(A) 140(B) 240 (C) 440 (D) 640 (E) 84016. Which of the following in equal to(A)(B) (C) 2 (D) (E) 617. In the eight-term sequence A, B, C, D, E, F, G , H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A + H?(A) 17(B) 18 (C) 25 (D) 26 (E) 4318. Circles A, B, and C each have radius 1. Circles A and B share one point oftangency. Circle C has a point of tangency with the midpoint of AB. What is the area inside Circle C but outside Circle A and Circle B? (A) 32π- (B) 2π (C) 2 (D) 34π (E) 12π+19. In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town’s popu lation during this twenty-year period?(A) 42(B) 47 (C) 52 (D) 57 (E) 6220. Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect? (A)16 (B) 15 (C) 14 (D) 13 (E) 1221. Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?(A) 711(B) 913(C) 1115(D) 1519(E) 151622. Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?(A) 2500 (B) 2880 (C) 3120 (D) 3250 (E) 375023. Seven students count from 1 to 1000 as follows:·Alice says all the numbers, except she skips the middle number in each consecutive group of thre e numbers. That is Alice says 1, 3, 4, 6, 7, 9, …, 997, 999, 1000.·Barbara says all of the numbers that Alice doesn’t say, except she also skips the middle number in each consecutive grope of three numbers.·Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers. ·Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.·Finally, George says the only number that no one else says.What number does George say?(A) 37 (B) 242 (C) 365 (D) 728 (E) 99824. Two distinct regular tetrahedra have all their vertices among the vertices of thesame unit cube. What is the volume of the region formed by the intersection of the tetrahedra?(A)112 (B) (C) (D) 16 (E)25. Let R be a square region and 4n an integer. A point X in the interior of R is called n-ray partitional 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) 25002011AMC10美国数学竞赛A 卷1. 某通讯公司手机每个月基本费为20美元, 每传送一则简讯收 5美分（一美元=100 美分）。

2011AMC10美国数学竞赛A卷附中文翻译和答案

2011AMC10美国数学竞赛A卷1. A cell phone plan costs $20 each month, plus 5¢ per text message sent, plus 10¢ for each minute used over 30 hours. In January Michelle sent 100 text messages and talked for 30.5 hours. How much did she have to pay?(A) $24.00(B) $24.50(C) $25.50(D) $28.00(E) $30.002. A small bottle of shampoo can hold 35 milliliters of shampoo, Whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?(A) 11(B) 12(C) 13(D) 14(E) 153. Suppose [a b] denotes the average of a and b, and {a b c} denotes the average of a, b, and c. What is {{1 1 0} [0 1] 0}?(A) 29(B)518(C)13(D) 718(E) 234. Let X and Y be the following sums of arithmetic sequences: X= 10 + 12 + 14 + …+ 100.Y= 12 + 14 + 16 + …+ 102.What is the value of Y X?(A) 92(B) 98(C) 100(D) 102(E) 1125. At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?(A) 12 (B) 373 (C) 887 (D) 13 (E) 146. Set A has 20 elements, and set B has 15 elements. What is the smallest possible number of elements in A ∪B, the union of A and B?(A) 5(B) 15 (C) 20 (D) 35 (E) 3007. Which of the following equations does NOT have a solution?(A)2(7)0x +=(B) -350x += (C) 20= (D)80= (E) -340x -=8. Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?(A) 20(B) 30 (C) 40 (D) 50 (E) 609. A rectangular region is bounded by the graphs of the equations y=a, y=-b, x=-c, and x=d, where a, b, c, and d are all positive numbers. Which of the following represents the area of this region?(A) ac + ad + bc + bd (B) ac – ad + bc – bd (C) ac + ad – bc – bd(D) –ac –ad + bc + bd (E) ac – ad – bc + bd10. A majority of the 20 students in Ms. Deameanor’s class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. 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) 7711. Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7·EB. What is the ratio of the area of EFGH to the area of ABCD?(A)4964 (B) 2532 (C) 78 (D) (E)12. The players on a basketball team made some three -point shots, some two -point shots, 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 was 61 points. How many free throws did they make?(A) 13(B) 14 (C) 15 (D) 16 (E) 1713. How many even integers are there between 200 and 700 whose digits are alldifferent and come from the set {1, 2, 5, 7, 8, 9}?(A) 12(B) 20 (C) 72 (D) 120 (E) 20014. A pair of standard 6-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) 51815. Roy bought a new battery -gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?(A) 140(B) 240 (C) 440 (D) 640 (E) 84016. Which of the following in equal to(A)(B) (C) 2 (D) (E) 617. In the eight -term sequence A, B, C, D, E, F, G, H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A + H?(A) 17(B) 18 (C) 25 (D) 26 (E) 4318. Circles A, B, and C each have radius 1. Circles A and B share one point oftangency. Circle C has a point of tangency with the midpoint of AB. What is the area inside Circle C but outside Circle A and Circle B? (A) 32π- (B) 2π (C) 2 (D) 34π (E) 12π+19. In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town’s population during this twenty -year period?(A) 42(B) 47 (C) 52 (D) 57 (E) 6220. Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect? (A)16 (B) 15 (C) 14 (D) 13 (E) 1221. Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?(A) 711(B) 913(C) 1115(D) 1519(E) 151622. Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?(A) 2500(B) 2880(C) 3120(D) 3250 (E) 375023. Seven students count from 1 to 1000 as follows:·Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is Alice says 1, 3, 4, 6, 7, 9, …, 997, 999, 1000.·Barbara says all of the numbers that Alice doesn’t say, except she also skips the middle number in each consecutive grope of three numbers.·Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.·Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.·Finally, George says the only number that no one else says.What number does George say?(A) 37(B) 242(C) 365(D) 728 (E) 99824. Two distinct regular tetrahedra have all their vertices among the vertices of thesame unit cube. What is the volume of the region formed by the intersection of the tetrahedra?(A)112 (B) (C) (D) 16 (E)25. Let R be a square region and 4n an integer. A point X in the interior of R is called n -ray partitional 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) 25002011AMC10美国数学竞赛A 卷1. 某通讯公司手机每个月基本费为20美元, 每传送一则简讯收 5美分（一美元=100 美分）。

2011年国际大学生数学竞赛试题解答第二天

1
Solution. (a) Let M be the set of males, F the set of females, and E the set of emales. Consider the (tripartite) graph G with vertices M ∪ F ∪ E and edges for likes. A 3-cycle is then a possible family. We’ll call G the graph of likes. First, let k = n 2 . Then n has to be even and we need to construct a graph of likes with no 3-cycles. We’ll do the following: divide each of the sets M , F , and E into two equal parts and draw all edges between two parts as shown below:
n b) Show that if k ≥ 34 , then it is always possible to create n disjoint married triples, thus marrying all of the expedition members.
Fedor Duzhin and Nick Gravin, Singapore
What are the possible values of lim xn ? Can such a sequence diverge? Johnson Olaleru, Lagos Solution 1. We prove by induction that 0 < 1 − xn < Then we will have (1 − xn ) → 0 and therefore xn → 1. The case n = 0 is true since

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。

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

A metric, Safety Probability has been defined as the probability that no frequency points conflicts existing among the users. The safety probabilities of the results obtained in our paper are all greater than 99.99% which is “safe” enough for the users.
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

2011美赛MCM赛题中英对照版[1]

2011 Contest ProblemsMCM PROBLEMS PROBLEM 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 devel op a “practical” course?A题：单板滑雪场地请设计一个单板滑雪场（现为“半管”或“U型池”）的形状，以便能使熟练的单板滑雪选手最大限度地产生垂直腾空。

“垂直腾空“是超出“半管”边缘以上的最大的垂直距离。

定制形状时要优化其他可能的要求，如：在空中产生最大的身体扭转。

在制定一个“实用”的场地时哪些权衡因素可能需要？PROBLEM B: Repeater Coordination B题中继站的协调The 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.甚高频无线电频谱包含信号的发送和接受。

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 中文AMC 10B Problem

2011 AMC 10B ProblemsProblem 1What is ?求的值Problem 2Josanna's test scores to date are and . Her goal is to raise here test average at least points with her next test. What is the minimum test score she would need to accomplish this goal?迄今为止乔萨娜的测验成绩是90,80,70,60和85.她的目标是连同下一次测验成绩，测验成绩的平均分至少要提高3分。

为达到目标，她要得到的测验成绩最少多少分？Problem 3At a store, when a length is reported as inches that means the length is at least inches and at most inches. Suppose the dimensions of a rectangular tile are reported as inches by inches. In square inches, what is the minimum area for the rectangle?商店标注的长度为x英寸的意思是长度在（x-0.5）英寸和（x+0.5）英寸之间。

设长方形瓷砖的尺寸标注尺寸为2英寸⨯3英寸。

以平方英寸计算，这种长方形的最小面积是多少？LeRoy and Bernardo went on a week-long trip together and agreed to share the costsequally. 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 dollars andBernardo had paid dollars, where . How many dollars must LeRoy give to Bernardo so that they share the costs equally?勒罗伊和伯纳多一起去为期一周的旅游，并且说好均摊费用。

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A题：滑雪场问题
请设计一个单板滑雪场（现为“半管”或“U型池”）的形状，以便能使熟练的单板滑雪选手最大限度地产生垂直腾空。

“垂直腾空“是超出“半管”边缘以上的最大的垂直距离。

定制形状时要优化其他可能的要求，如：在空中产生最大的身体扭曲。

在制定一个“实用”的场地时哪些权衡因素可能需要？
微分方程，确定一个形状，现在是半圆形的
B题中继站的协调
甚高频无线电频谱包含信号的发送和接收。

这种限制是可以被“中继站”克服，中继站捕捉到弱信号,放大它们,再用不同的频率重新发送。

这样, 低功耗的用户们/站（例如移动电话用户/站）在用户与用户（站与站）之间无法直接接触联系的情况下，通过使用中继站,就可以相互交流。

然而,中继站之间会相互干扰,除非它们离的足够远或者它们通过充分分离的频率来传送。

除了地理的分离、“持续的音频编码控制系统”,有时被称为“私人专线”技术，可以用来减轻干扰问题。

该系统给每一个中继站连接了一个独立的次声频音,这个次声频音由想通过中继站交流的每一个用户所发送。

中继站只回应接收到的特殊的私人专线信号。

通过这个系统,两个邻近的中继站可以共享相同的频率对(包括接收和发送);这样，在一个特定的区域可以容纳更多的中继站(并因此能同时容纳更多的用户)。

单纯形方法，分配问题：匈牙利方法，
问题：
1.为一个半径40英里的圆形平台区域决定最少数量的中继站，设计一个方案，要求能容纳1000个用户同时在线。

假设甚高频无线电频谱范围是145兆赫~148兆赫,中继站发送的频率要么是600千赫以上，要么低于接收的频率600千赫，共有54种不同的私人专线可用。

2.如果有10,000用户同时被容纳，如何改变你的解决方案?
3. 在由于山区引起信号传播阻碍的地区，讨论这样的情形。

频率和距离
甚高频无线频谱包括视距发送和接收。

这种限制可以被“中继器”所克服，中继器接收到微弱的信号，放大然后在不同的频率重传信号。

所以，如果使用一个中继器，低功率用户（比如移动台）可以和另外一个不能直接建立用户至用户联系的用户进行通信。

但是，如果中继器之间距离不够远或者频率相距不够大，中继器之间会产生相互的干扰。

除了空间上的远离外，“连续语音控制静噪系统”（CTCSS），有时被称为“专线”(PL)的技术可以被用来解决干扰的问题。

这个系统以不同的亚音频和每个中继器联系，所有希望通过那个中继器通信的用户都发送那个亚音频。

只有当以自己的专线音频接收信号时中继器才响应。

有了这个系统，两个相邻的中继器可以共享同样的频率对（为了接收和发送）；所以更多地中继器（也意味着更多的用户）可以共存于一个特别的地带。

对于一个半径为40英里的圆形平坦地带，决定满足1000个同时存在的用户的最小中继器数目。

假设可用频谱从145到148MHz，中继器的发送频率高于或低于接收频率600kHz，则有54个不同的亚音频可供使用。

璇姐(397449389) 13:49:45
甚高频无线频谱包括视距发送和接收。

这种限制可以被“中继器”所克服，中继器接收到微弱的信号，放大然后在不同的频率重传信号。

所以，如果使用一个中继器，低功率用户（比如移动台）可以和另外一个不能直接建立用户至用户联系的用户进行通信。

但是，如果中继器之间距离不够远或者频率相距不够大，中继器之间会产生相互的干扰。

除了空间上的远离外，“连续语音控制静噪系统”（CTCSS），有时被称为“专线”(PL)的技术可以被用来解决干扰的问题。

这个系统以不同的亚音频和每个中继器联系，所有希望通过那个中继器通信的用户都发送那个亚音频。

只有当以自己的专线音频接收信号时中继器才响应。

对于一个半径为40英里的圆形平坦地带，决定满足1000个同时存在的用户的最小中继器数目。

假设可用频谱从145到148MHz，中继器的发送频率高于或低于接收频率600kHz，则有54个不同的亚音频可供使用。

中继台通讯距离的工程计算
1．无线电波传输损耗工程实用公
式LM(dB)=88.1+20lgF-20lgh1h2+40lgd 式中：F—通讯工作频率(MHz) h1—通讯对象A点天线高度(m) h2—通讯对象B点天线高度(m) d—A点和B点的通讯距离(m) 上述实用公式仅限于VHF 150MHz和UHF 400～470MHz频段，并且地形起伏高度在15m左右，通讯距离在65km范围内。

2．系统无线设备通讯距离的计算(以下举例说明) (1)假设已知条件 a.系统工作频率：TX 465MHz RX 455MHz b.中继台参数和架设数据：发射功率：20W (43dBm) 接收灵敏度：-116dBm 同轴电缆损耗：2dB(1/2〃馈管40m长、5dB/100m) 全向天线增益：9.8dbi 天线架设高度：30m c.对讲机参数发射功率：4W(36dBm) 接收灵敏度：-116dBm 对讲机天线增益：0dBi 对讲机高度：1.5m (2)中继台与对讲机的系统增益在本例中，所谓系统增益就是对讲机发射信号给中继台接收机允许衰减的最大值，若不考虑电缆损耗和天线增益的条件下：系统增益(dB)=发射功率(dBm)-接收灵敏度(dBm) 若考虑电缆损耗、天线增益的条件下,本例系统增益
为：SG(dB)=Pt+PA-(RA+CL+RR) =36+0-(9.8-2-116) =144.2(dB)式中：Pt——对讲机发射功率PA——对讲机天线增益RA——中继台天线增益CL——同轴电缆损耗RR——中继台接收灵敏度(3)如果系统增益等于电波传输的损耗，则说明通讯距离的电波能量已达极限，若系统增益小于传输损耗则表明通讯可能建立不起来。

将系统增益代入电波传输损耗工程公
式：144.2=88.1+201g455-201g1.5×30+401gd144.2=88.1+53.2-33+ 401gd 35.9=401gd d=7.9km (4)上式仅计算了上行信号(对讲机发给中继台)可通讯的保守距离，而未计算下行信号(中继台发给对讲机)可覆盖的距离，通常由于中继台发射功率较大，其下行信号往往优于上行信号的通讯距离。

由于系统通话是双向的，因此系统的保守通话距离往往以上行信号为准来计算。

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