美国数学建模MCM_2005_题目

AB DC B 2005年 美国AMC8 (2005年11月 日 时间40分钟)1. 小妮将一数乘以2得到60。

不过，她应当将这个数除以2才会得到正确的答案。

正确的答案应该是多少？ (A) 7.5 (B) 15 (C) 30 (D) 120 (E) 240。

2. 卡尔某天在文具店买了每个美金$2.5元的资料夹5个。

这个文具店在卡尔买后的隔天降价20%。

假如卡尔多等一天才买，问他可以节省多少元？(A) $1.00 (B) $2.00 (C) $2.50 (D) $2.75 (E) $5.00。

3. 至少要将下图正方形ABCD 中多少个空白的小正方形涂黑后，才可以使得着色后的图形是对称于对角线BD ？ (A) 1 (B) 2 (C) 3(D) 4 (E) 5。

4. 一个正方形与一个三角形的周长相等。

三角形的三边长分别为6.1公分、8.2公分及9.7公分。

此正方形的面积是多少平方公分？ (A) 24 (B) 25 (C) 36 (D) 48 (E) 64。

5. 汽水有以6罐一捆、12罐一捆及24罐一捆三种包装方式出售。

至少要买多少捆才会刚好买到90罐汽水？ (A) 4 (B) 5 (C) 6 (D) 8 (E) 15。

6. 设d 是一位数字。

有多少个数值d 可以使得2.00d 5>2.005？(A) 0 (B) 4 (C) 5 (D) 6 (E) 10。

7. 小明先往南走了21公里，再往东走了43公里，最后再往南走了21公里。

小明从出发点到最后停止点的直线距离是多少公里？ (A) 1 (B) 411 (C) 211 (D) 431 (E) 2。

8. 设m 及n 为正奇数，下列哪一个也是奇数？(A) m +3n (B) 3m -n (C) 3m 2+3n 2 (D) (nm +3)2 (E) 3mn 。

9. 在四边形ABCD 中，边AB 及BC 的长均为10，边CD 及DA 的长均为17，且角∠ADC 为60︒。

美国数学竞赛AMC8 – 2005年真题解析(英文解析+中文解析)Problem 1Answer: BSolution:If x is the number, then 2x=60 and x=30. Dividing the number by 2 yields 15.中文解析：按照Connie的计算，这个数乘以2是60，可知这个数是30. 应该做的计算是30除以2，因而正确答案应该是15. 答案是B。

Problem 2Answer: CSolution:Karl paid 5*2.5=$12.5. 20% of this cost that he saved is 12.5*0.2=$2.5.中文解析：Karl按原价买了5个文件夹，支付的费用是：2.5*5=12.5. 折扣价是：1.25*0.8=10。

如果Karl 等一天，可以省2.5元。

答案是C.Problem 3Answer: DSolution:Rotating square ABCD counterclockwise 45° so that the line of symmetry BD is a vertical line makes it easier to see that 4 squares need to be colored to match its corresponding square.中文解析：如上图所示，以BD为对称轴，标蓝色的方块需要涂黑。

共4块，答案是D。

Problem 4Answer: CSolution:The perimeter of the triangle is 6.1+8.2+9.7=24cm. A square's perimeter is four times its side length, since all its side lengths are equal. If the square's perimeter is 24, the side length is24/4=6, and the area is 6*6=36.中文解析：三角形的周长是：6.1+8.2+9.7=24. 正方形的周长和三角形相等，也是24，则其边长是24/4=6. 其面积是：6*6=36. 答案是C。

2005数学建模试题1．（10分）设某产品的供给函数)(p ϕ与需求函数)(p f 皆为线性函数： 78)(65)(+-=+=p p f p p ϕ其中p 为商品单价，试判断市场是否稳定并给出推理过程。

2．（10分）某植物园的植物基因型为AA 、Aa 、aa ，人们计划用AA 型植 物与每种基因型植物相结合的方案培育后代（遗传方式为常染色体遗传），经过若干代后，这种植物后代的三种基因型分布将出现什么情形？总体趋势如何？ 3．（10分）建立捕鱼问题的模型，并通过求解微分方程的办法给出最大的 捕捞量。

4. （10分）试建立Lanchester 游击战模型，并在无自然损失及没有增援的条件下求解模型，给出敌对双方获胜的条件。

5. （10分）根据水情资料， 某地汛期出现平水水情的概率为0.7, 出现高 水水情的概率为0.2, 出现洪水水情的概率为0.1。

.位于江边的某工地对其大型施工设备拟定三个处置方案：a) 运走，需支付运费20万元。

b) 修堤坝保护，需支付修坝费8万元。

c) 不作任何防范，不需任何支出。

若采用方案（1），那么无论出现任何水情都不会遭受损失；若采用方案（2），则仅当发生洪水时，因堤坝冲垮而损失600万元的设备；若采用方案（3），那么当出现平水水位时不遭受损失，发生高水水位时损失部分设备而损失300万元，发生洪水时损失设备600万元。

根据上述条件，选择最佳决策方案。

6．（10分）由七种规格的包装箱要装到两辆铁路平板车上去。

包装箱的宽和高时一样的，但厚度（t ，以厘米计）及重量（ω，以公斤计）是不同的。

下表给出了每种包装箱的厚度、重量以及数量。

每辆平板车有10.2米的地方可用来装包装箱（像面包片那样），载重为40吨。

由于当地货运得限制，对C 5，C 6，C 7类的包装箱的总数有一个特别的限制：这类箱子所占的空间（厚度）不能超过302.7厘米。

试把包装箱（见下表）装到平板车上去使得浪费的空间最小。

马剑整理历年美国大学生数学建模赛题目录MCM85问题-A 动物群体的管理 (3)MCM85问题-B 战购物资储备的管理 (3)MCM86问题-A 水道测量数据 (4)MCM86问题-B 应急设施的位置 (4)MCM87问题-A 盐的存贮 (5)MCM87问题-B 停车场 (5)MCM88问题-A 确定毒品走私船的位置 (5)MCM88问题-B 两辆铁路平板车的装货问题 (6)MCM89问题-A 蠓的分类 (6)MCM89问题-B 飞机排队 (6)MCM90-A 药物在脑内的分布 (6)MCM90问题-B 扫雪问题 (7)MCM91问题-B 通讯网络的极小生成树 (7)MCM 91问题-A 估计水塔的水流量 (7)MCM92问题-A 空中交通控制雷达的功率问题 (7)MCM 92问题-B 应急电力修复系统的修复计划 (7)MCM93问题-A 加速餐厅剩菜堆肥的生成 (8)MCM93问题-B 倒煤台的操作方案 (8)MCM94问题-A 住宅的保温 (9)MCM 94问题-B 计算机网络的最短传输时间 (9)MCM-95问题-A 单一螺旋线 (10)MCM95题-B A1uacha Balaclava学院 (10)MCM96问题-A 噪音场中潜艇的探测 (11)MCM96问题-B 竞赛评判问题 (11)MCM97问题-A Velociraptor(疾走龙属)问题 (11)MCM97问题-B为取得富有成果的讨论怎样搭配与会成员 (12)MCM98问题-A 磁共振成像扫描仪 (12)MCM98问题-B 成绩给分的通胀 (13)MCM99问题-A 大碰撞 (13)MCM99问题-B “非法”聚会 (14)MCM2000问题-A空间交通管制 (14)MCM2000问题-B: 无线电信道分配 (14)MCM2001问题- A: 选择自行车车轮 (15)MCM2001问题-B 逃避飓风怒吼（一场恶风...） .. (15)MCM2001问题-C我们的水系-不确定的前景 (16)MCM2002问题-A风和喷水池 (16)MCM2002问题-B航空公司超员订票 (16)MCM2002问题-C (16)MCM2003问题-A: 特技演员 (18)MCM2003问题-B: Gamma刀治疗方案 (18)MCM2003问题-C航空行李的扫描对策 (19)MCM2004问题-A：指纹是独一无二的吗？ (19)MCM2004问题-B：更快的快通系统 (19)MCM2004问题-C安全与否？ (19)MCM2005问题A.水灾计划 (19)MCM2005B.Tollbooths (19)MCM2005问题C：不可再生的资源 (20)MCM2006问题A: 用于灌溉的自动洒水器的安置和移动调度 (20)MCM2006问题B: 通过机场的轮椅 (20)MCM2006问题C : 抗击艾滋病的协调 (21)MCM2007问题B ：飞机就座问题 (24)MCM2007问题C：器官移植：肾交换问题 (24)MCM2008问题A:给大陆洗个澡 (28)MCM2008问题B：建立数独拼图游戏 (28)MCM85问题-A 动物群体的管理在一个资源有限，即有限的食物、空间、水等等的环境里发现天然存在的动物群体。

目录2001 MCM A: Choosing a Bicycle Wheel (2)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...) (3)2002 MCM A: Wind and Waterspray (5)2002 MCM B: Airline Overbooking (5)2003 MCM A: The Stunt Person (6)2003 MCM B: Gamma Knife Treatment Planning (7)2004 MCM A: Are Fingerprints Unique? (8)2004 MCM B: A Faster QuickPass System (8)2005 MCM A: Flood Planning (9)2005 MCM B: Tollbooths (9)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation (10)2006 MCM B: Wheel Chair Access at Airports (10)2007 MCM A: Gerrymandering (12)2007 MCM B: The Airplane Seating Problem (12)2008 MCM A: Take a Bath (13)2008 MCM B: Creating Sudoku Puzzles (13)2009 MCM A: Designing a Traffic Circle (13)2009 MCM B: Energy and the Cell Phone (14)2001 MCM A: Choosing a Bicycle WheelCyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike.Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course.Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right.The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear.Task 1. Provide a table giving the wind speed at which the power required for a solid rear wheel is less than for a spoked rear wheel.The table should include the wind speeds for different road grades starting from zero percent to ten percent in one percent increments.(Road grade is defined to be the ratio of the total rise of a hill divided 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 rider will lose about 8 kph for a five percent grade over 100meters.∙Task 2. Provide an example of how the table could be used for a specific time trial course.∙Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision.2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...)Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare.The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned.A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of the coastal strip. (More accurate data, if sought, are widely available.)The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees (including some coming from farther north by other routes), so some traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only very slowly. Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed:1.Under what conditions does the plan for turning the twocoastal-bound lanes of I-26 into two lanes of Columbia-boundtraffic, essentially turning the entire I-26 into one-way traffic, significantly improve evacuation traffic flow?2.In 1999, the simultaneous evacuation of the state's entire coastalregion was ordered. Would the evacuation traffic flow improve under an alternative strategy that staggers the evacuation, perhapscounty-by-county over some time period consistent with the pattern of how hurricanes affect the coast?3.Several smaller highways besides I-26 extend inland from the coast.Under what conditions would it improve evacuation flow to turnaround traffic on these?4.What effect would it have on evacuation flow to establish moretemporary shelters in Columbia, to reduce the traffic leavingColumbia?5.In 1999, many families leaving the coast brought along their boats,campers, and motor homes. Many drove all of their cars. Under what conditions should there be restrictions on vehicle types or numbers of vehicles brought in order to guarantee timely evacuation?6.It has been suggested that in 1999 some of the coastal residentsof Georgia and Florida, who were fleeing the earlier predictedlandfalls of Hurricane Floyd to the south, came up I-95 andcompounded the traffic problems. How big an impact can they have on the evacuation traffic flow? 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.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 MCM A: Wind and WatersprayAn ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area.Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.2002 MCM B: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations.Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.Consider the overbooking issue in light of the current situation: Less flights by airlines from point A to point B Heightened security at and around airports Passengers' fear Loss of billions of dollars in revenue by airlines to dateBuild a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company's revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling “bumped”passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.2003 MCM A: The Stunt PersonAn exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! A stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by camera, etc.).Your job is to:∙determine what size boxes to use∙determine how many boxes to use∙determine how the boxes will be stacked∙determine if any modifications to the boxes would help∙generalize to different combined weights (stunt person & motorcycle) and different jump heightsNote that, in “Tomorrow Never Dies”, the James Bond character on a motorcycle jumps over a helicopter.2003 MCM B: Gamma Knife Treatment PlanningStereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators.The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. All 201 beams simultaneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed a “shot.” Shots can be represented as different spheres. Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume is a bounded, three-dimensional digital image that usually consists of millions of points.The goal of radiosurgery is to deplete tumor cells while preserving normal structures. Since there are physical limitations and biological uncertainties involved in this therapy process, a treatment plan needs to account for all those limitations and uncertainties. In general, an optimal treatment plan is designed to meet the following requirements.1.Minimize the dose gradient across the target volume.2.Match specified isodose contours to the target volumes.3.Match specified dose-volume constraints of the target and criticalorgan.4.Minimize the integral dose to the entire volume of normal tissuesor organs.5.Constrain dose to specified normal tissue points below tolerancedoses.6.Minimize the maximum dose to critical volumes.In gamma unit treatment planning, we have the following constraints:1.Prohibit shots from protruding outside the target.2.Prohibit shots from overlapping (to avoid hot spots).3.Cover the target volume with effective dosage as much as possible.But at least 90% of the target volume must be covered by shots.e as few shots as possible.Your tasks are to formulate the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution. While designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient.2004 MCM 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.2004 MCM 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 A: Flood PlanningLake Murray in central South Carolina is formed by a large earthen dam, which was completed in 1930 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?2005 MCM 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 wherethere 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 A: Positioning and Moving Sprinkler Systems for IrrigationThere 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 system 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. You should determine the number of sprinklers and the spacing between sprinklers, and you should find a schedule 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 put 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 possible.2006 MCM 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 arenot 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 A: GerrymanderingGerrymandering The 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 “unnatural” 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 York.2007 MCM B: The Airplane Seating 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.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:/2006/11/14/business/14boarding.html2008 MCM A: Take a BathConsider 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.2008 MCM B: Creating 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 A: Designing a Traffic CircleMany cities and communities have traffic circles—from large ones with many lanes in the circle (such as at the Arc de Triomphe in Paris and the Victory Monument in Bangkok) to small ones with one or two lanes in the circle. Some of these traffic circles position a stop sign or a yield sign on every incoming road that gives priority to traffic already in the circle; some position a yield sign in the circle at each incoming road to give priority to incoming traffic; and some position a traffic light on eachincoming road (with no right turn allowed on a red light). Other designs may also be possible.The goal of this problem is to use a model to determine how best to control traffic flow in, around, and out of a circle. State clearly the objective(s) you use in your model for making the optimal choice as well as the factors that affect this choice. Include a Technical Summary of not more than two double-spaced pages that explains to a Traffic Engineer how to use your model to help choose the appropriate flow-control method for any specific traffic circle. That is, summarize the conditions under which each type of traffic-control method should be used. When traffic lights are recommended, explain a method for determining how many seconds each light should remain green (which may vary according to the time of day and other factors). Illustrate how your model works with specific examples.2009 MCM B: Energy and the Cell PhoneThis question involves the “energy” consequences of the cell phone revolution. Cell phone usage is mushrooming, and many people are using cell phones and giving up their landline telephones. What is the consequence of this in terms of electricity use? Every cell phone comes with a battery and a recharger.Requirement 1Consider the current US, a country of about 300 million people. Estimate from available data the number H of households, with m members each, that in the past were serviced by landlines. Now, suppose that all the landlines are replaced by cell phones; that is, each of the m members of the household has a cell phone. Model the consequences of this change for electricity utilization in the current US, both during the transition and during the steady state. The analysis should take into account the need for charging the batteries of the cell phones, as well as the fact that cell phones do not last as long as landline phones (for example, the cell phones get lost and break).Requirement 2Consider a second “Pseudo US”—a country of about 300 million people with about the same economic status as the current US. However, this emerging country has neither landlines nor cell phones. What is the optimal way of providing phone service to this country from an energy perspective? Of course, cell phones have many social consequences and uses that landline phones do not allow. A discussion of the broad and hidden。

2005 AMC12中文试题及答案56th AMC 12 A 20051. x 的10%和y 的20%都是2. 试问x y -之值为何？(A) 1 (B) 2 (C) 5 (D) 10 (E) 202. 方程式273x +=及102bx -=-有相同的解x . 试问b 之值为何？(A) 8- (B) 4- (C) 2- (D) 4 (E) 83. 有一长方形长是宽的两倍, 其一对角线的长度为x . 试问此长方形的面积为多少？ (A) 214x (B) 225x (C) 212x (D) 2x (E) 232x4. 某商店通常每扇窗户卖美金$100元. 本周此商店每买四扇窗户就免费送一扇窗户. 戴维需要七扇窗户, 道奇需要八扇窗户. 试问他们两人合起来买要比分开来买节省美金多少元？(A) 100 (B) 200 (C) 300 (D) 400 (E) 5005. 有20个数的平均数为30, 且另外30个数的平均数为20. 试问这50个数的平均数是多少？(A) 23 (B) 24 (C) 25 (D) 26 (E) 276. 乔许的家与麦克的家相距13公里. 昨天乔许先骑着他的脚踏车朝麦克家去, 过了一会儿麦克才骑着他的脚踏车朝乔许家去. 当他们相遇时, 乔许骑车的时间为麦克的两倍, 而乔许骑车的速率为麦克的45. 试问他们相遇时麦克已经骑了多少公里？(A) 4 (B) 5 (C) 6 (D) 7 (E) 87. 如图所示, 正方形ABCD的边AB50E介于B与H之间, 且1BE=. 试问内部正方形EFGH的面积为多少？(A) 25(B) 32(C) 36(D) 40(E) 428. 设A、M、C都是一位数字满足：(10010)()2005++++=. 试问A M C A M CA之值为何？(A) 1(B) 2(C) 3(D) 4(E) 59. 使得方程式2x ax x+++=恰仅有一个x解的a有两个值. 试问a这两个值的4890和是多少？(A) 16-(B) 8-(C) 0(D) 8(E) 2010. 一个边长为n单位的木头正立方体, 将它的六个面都涂成红色, 并将它切割成3n个单位正立方体. 若这些单位正立方体的总面数有四分之一是红色的面, 则n是多少？(A) 3(B) 4(C) 5(D) 6(E) 711. 有多少个三位数满足：十位数字是百位数字与个位数字的平均数？(A) 41(B) 42(C) 43(D) 44(E) 4512. 给定(1,1)B两点. 试问在线段AB上除A、B两点外, 有多少A、(100,1000)个点它的两个坐标都是整数？(A) 0(B) 2(C) 3(D) 8(E) 913. 如图所示的五星形, 在英文字母A 、B 、C 、D 、E 处填入数字3、5、6、7、9 (不一定按此顺序). 在各线段AB 、BC 、CD 、DE 、EA (也不一定按此顺序)两端数字的和恰可排成等差数列. 试问此等差数列正中间那一项的数为何？(A) 9 (B) 10 (C) 11 (D) 12(E) 1314. 随机地先将一个公正骰子上的一点抹掉, 每个点被抹掉的机会相同, 然后投掷这个骰子. 试问这个骰子朝上那个面出现奇数点的机率是多少？ (A) 511 (B) 1021 (C) 12 (D) 1121 (E) 61115. 设AB 为圆的直径, C 为AB 上的一点使得2AC BC =, 并设D 与E 为圆周上的两点使得DC AB ⊥且DE 为圆的另一直径. 试问DCE ∆面积与ABD ∆面积的比值为何？(A)16 (B) 14 (C) 13 (D) 12 (E) 2316. 三个半径为s的圆画在xy平面的第一象限. 第一个圆与x轴、y轴相切, 第二个圆与第一个圆及x轴相切, 第三个圆与第一个圆及y轴相切. 另一个半径为r (r s)的圆与两轴、第二个圆及第三个圆相切. 试问rs之值为何？(A) 5(B) 6 (C) 8(D) 9(E) 1017. 一个单位正立方体被切两刀形成三个三角柱, 其中有两个三角柱全等, 如(图一)所示. 再将此正立方体以相同的方式沿虚线再切两刀, 如(图二)所示, 使得正立方体被切成九小块. 试问包含顶点W的那一小块的体积为何？(图一) (图二)(A)112(B)19(C)18(D)16(E)1418. 称一个数是「质样数」, 如果此数为合成数, 但不能被2、3或5整除. 最小的三个「质样数」为49、77及91. 小于1000的质数有168个. 试问小于1000的「质样数」有多少个？(A) 100(B) 102(C) 104(D) 106(E) 10819. 一个有缺陷的汽车里程表永远由3直接跳到5, 即不论是在哪一位数永远会将数字4略过. 例如：此里程表在000039行驶一公里后会直接跳到000050. 若里程表上显现的是002005, 则汽车实际行驶的距离为多少公里？(A) 1404 (B) 1462 (C) 1604 (D) 1605 (E) 180420. 对于每一个在[0,1]中的x , 定义12, 02()122, 1.2x x f x x x ⎧≤≤⎪⎪=⎨⎪-<≤⎪⎩若 若 对所有的正整数2n ≥, 令[2]()(())f x f f x =, [1][]()(())n n f x f f x +=.试问在[0,1]中有多少个x 可使得[2005]1()2f x =？ (A) 0 (B) 2005 (C) 4010 (D) 22005 (E) 2005221. 若2a ≥, 1b ≥且0c ≥为整数. 试问2005log a b c =且2005a b c ++=有多少组解？(A) 0 (B) 1 (C) 2 (D) 3 (E)422. 一个长方体P 内接于一个半径为r 的球内, 已知P 的表面积为384, 且12条棱长之和为112. 试问r 为多少？(A) 8 (B) 10 (C) 12 (D) 14 (E) 1623. 从集合2325{2,2,2,,2}L 中任意选取两个不同的数a 及b . 试问log a b 为整数的机率是多少？ (A) 225 (B) 31300 (C) 13100 (D) 750 (E) 1224. 设()(1)(2)(3)Q x可以找到三次多项式=---. 试问有多少个多项式()P x x x x=⋅？P Q x P x R x()R x使得(())()()(A) 19 (B) 22 (C) 24 (D) 27 (E) 3225. 令S为所有的点其坐标为(,,)x y z, 其中x, y, z都选自集合{0,1,2}. 试问有多少个正三角形其顶点都在S内？(A) 72(B) 76(C) 80 (D) 84(E) 88答案：1 ( D )2 ( B )3 ( B )4 ( A )5 ( B )6 ( B )7 ( C )8 ( D )9 ( A ) 10 ( B )11 ( E ) 12 ( D ) 13 ( D ) 14 ( D ) 15 ( C )16 ( D ) 17 ( A ) 18 ( A ) 19 ( B ) 20 ( E )21 ( C ) 22 ( B ) 23 ( B ) 24 ( B ) 25 ( C )。

AMC10美国数学竞赛真题2005B卷A scout troop buys candy bars at a price of five for $. They sell all the candy bars at a price of two for $. What was the profit, in dollars?SolutionProblem 2A positive number has the property that of is . What is ?SolutionProblem 3A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?SolutionProblem 4For real numbers and , define . What is the value ofSolutionProblem 5Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?SolutionAt the beginning of the school year, Lisa's goal was to earn an A on at leastof her quizzes for the year. She earned an A on of the first quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?SolutionProblem 7A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?SolutionProblem 8An -foot by -foot ?oor is tiled with square tiles of size foot by foot. Each tile has a pattern consisting of four white quarter circles of radius foot centered at each corner of the tile. The remaining portion of the tile is shaded.How many square feet of the ?oor are shaded?SolutionProblem 9One fair die has faces , , , , , and another has faces , , , , , . The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?SolutionIn , we have and . Suppose that is a point on line such that lies between and and . What is ?SolutionProblem 11The first term of a sequence is . Each succeeding term is the sum of the cubes of the digits of the previous term. What is the term of the sequence?SolutionProblem 12Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?SolutionProblem 13How many numbers between and are integer multiples of or but not ?SolutionProblem 14Equilateral has side length , is the midpoint of , and is the midpoint of . What is the area of ?SolutionProblem 15An envelope contains eight bills: ones, fives, tens, and twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $or more?SolutionProblem 16The quadratic equation has roots that are twice those of, and none of , , and is zero. What is the value of ?SolutionProblem 17Suppose that , , , and . What is ?SolutionProblem 18All of David's telephone numbers have the form , where , , , , , , and are distinct digits and in increasing order, and none is either or . How many different telephone numbers can David have?SolutionProblem 19On a certain math exam, of the students got points, got points, got points, got points, and the rest got points. What is the difference between the mean and the median score on this exam?SolutionProblem 20What is the average (mean) of all -digit numbers that can be formed by using each of the digits , , , , and exactly once?SolutionProblem 21Forty slips are placed into a hat, each bearing a number , , , , , , , , , or , with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let be the probability that all four slips bear the same number. Let be the probability that two of the slips bear a numberand the other two bear a number . What is the value of ?SolutionProblem 22For how many positive integers less than or equal to is evenly divisible by ?SolutionProblem 23In trapezoid we have parallel to , as the midpoint of , and as the midpoint of . The area of is twice the area of. What is ?SolutionProblem 24Let and be two-digit integers such that is obtained by reversing the digits of . The integers and satisfy for some positive integer . What is ?SolutionProblem 25A subset of the set of integers from to , inclusive, has the property that no two elements of sum to . What is the maximum possible number of elements in ?。

2005A.水灾计划
南卡罗来纳州中部的磨累河是由北部的一个巨大水坝形成的，这是在1930年为了发电而修建的，模拟一起洪水淹没下游的事件，这起事件是由于一次灾难性的地震损毁了水坝造成的。

两个问题：
Rawls Creek是水坝下游流入Saluda河的一条终年流动的河流，则当水坝损毁后在Rawls Creek将会出现多大的洪流，洪水的波及面将有多大？
S.C.国会大厦大楼在一座小山上，在S.C.国会大厦大楼能俯视Congaree 河。

洪水能如此巨大顺流以致于水将扩展到S.C.国会大厦大楼吗？
2005B.Tollbooths(收费亭)
像Garden State Parkway，Interstate 95等等这样的长途收费公路，通常是多行道的，被分成几条高速公路，在这些高速公路上每隔一定的间隔会设立一个通行税收费广场。

因为征收通行税通常不受欢迎，所以
应该尽量减少通过通行税收费广场引起的交通混乱给汽车司机带来的烦恼。

通常，收费亭的数量要多于进入收费广场的道路的数量。

进入通行税收费广场的时候，流到大量收费亭的车辆呈扇形展开，当离开通行税收费广场的时候，车流将只能按照收费广场前行车道路的数量排队按次序通过！从而，当交通是拥挤的时，拥挤在违背通行税广场上增加。

当交通非常拥挤的时候，因为每车辆付通行费的时间要求，阻塞也会出现在通行税收费广场入口处。

建立一个模型来确定在一个容易造成阻塞的通行税收费广场中应该部署的最优的收费亭的数量。

需要保证每一个进入收费广场的交通线路上都仅有一个收费亭。

与当今的实践相比较，在什么条件下这或多或少有效？
注意："最佳"的定义由你自己决定。

第20卷第4期2006年7月常熟理工学院学报Journal of Changshu Institute of T echnologyV ol.20N o.4July.2006浅析2005数学建模竞赛C题Ξ金　健,朱惠健(常熟理工学院数学系,江苏常熟　215500)摘　要:探讨了2005数学建模竞赛C题“雨量预报方法的评价”的命题意图,结合我校学生的竞赛论文分析了某些参赛队解题中的典型错误,并提供了用散乱数据的曲面拟合求解本题的一些关键要点。

关键词:Shepard插值;散乱数据插值;曲面拟合中图分类号:O141.4 文献标识码:A 文章编号:1008-2794(2006)04-0041-05 散乱数据插值(scattered data interpolation)主要研究根据给定散乱数据点构造光滑曲面的理论与方法,其历史可追溯到二十世纪二十年代,目前散乱数据插值技术已广泛应用于各类科学研究和工程技术中,如气象、勘探、医学、环保、可视化以及测量造型等,散乱数据插值问题的提法是:给定数据(X i,f i)∈R n R,i= 1,2,…,N,要求构造函数G=G(X)插值给定的函数值,即G(X i)=f i,i=1,2,…,N,插值方法的优劣可以从精确性、视觉效果、对参数的灵敏性、执行时间、存储量要求和在计算机上实现的简易性等方面进行比较[1],使用何种插值方法需视问题的实际情况而定。

1　“雨量预报评价”的插值模型1.1　题目雨量预报对农业生产和城市工作和生活有重要作用,但准确、及时地对雨量作出预报是一个十分困难的问题,广受世界各国关注。

我国某地气象台和气象研究所正在研究6小时雨量预报方法,即每天晚上20点预报从21点开始的4个时段(21点至次日3点,次日3点至9点,9点至15点,15点至21点)在某些位置的雨量,这些位置位于东经120度、北纬32度附近的53×47的等距网格点上。

同时设立91个观测站点实测这些时段的实际雨量,由于各种条件的限制,站点的设置是不均匀的。