2009-2010年度美国大联盟杯竞赛试题

A题设计一个交通环岛在许多城市和社区都建立有交通环岛，既有多条行车道的大型环岛（例如巴黎的凯旋门和曼谷的胜利纪念碑路口），又有一至两条行车道的小型环岛。

有些环岛在进入口设有“停车”标志或者让行标志，其目的是给已驶入环岛的车辆提供行车优先权；而在一些环岛的进入口的逆向一侧设立的让行标志是为了向即将驶入环岛的车辆提供行车优先权；还有一些环岛会在入口处设立交通灯（红灯会禁止车辆右转）；也可能会有其他的设计方案。

这一设计的目的在于利用一个模型来决定如何最优地控制环岛内部，周围以及外部的交通流。

该设计的目的在于可利用模型做出最佳的方案选择以及分析影响选择的众多因素。

解决方案中需要包括一个不超过两页纸，双倍行距打印的技术摘要，它可以指导交通工程师利用你们模型对任何特殊的环岛进行适当的流量控制。

该模型可以总结出在何种情况之下运用哪一种交通控制法为最优。

当考虑使用红绿灯的时候，给出一个绿灯的时长的控制方法（根据每日具体时间以及其他因素进行协调）。

找一些特殊案例，展示你的模型的实用性。

B题能源和手机这个问题涉及到手机革命的能源问题。

手机使用率迅速增加，许多人使用手机并放弃了固定电话。

这方面的电能使用会带来什么后果？每个手机都配备了电池和充电器。

要求1考虑现在的美国，人口约为3亿，从现有数据估计美国有H个家庭，每个家庭有M个成员，以前是使用固定电话的。

现在，假设所有的座机被手机取代，也就是说每个家庭成员都有一部手机。

建立当前美国在手机使用的过渡和稳定两个阶段用电改变的模型，分析应该考虑到对移动电话充电的需要，同时移动电话不能像固定电话那样长期使用也是一个现实问题（比如说移动电话可能会丢失或者损坏）要求2考虑“伪美国”--一个约3亿人口，跟当前美国具有相同的经济状况的国家。

然而，这个新兴国家既没有固定电话也没有移动电话，从能源角度看，为这个国家提供电话服务的最佳方式是什么？当然，手机有很多固定电话所不具有的用途和社会影响。

12. If x is doubled and then increased by 7, the result is 81. What isthe value of x?13. When the American Idol competition was down to the finaltwo contestants, the winner was ahead of the second placefinisher by 12 million votes. If the total number of votes was90 million, how many votes did the winner get?14. What is the sum of the values of a that satisfy the equation:(3)52 – 4(5 – a)2 ÷ 3 = 63?15. When each edge of a cube is increased by 50%, by whatpercent is the surface area of the cube increased?16. What is the sum of three consecutive even integers if the sumof the first and third integers is 128?17. Erin is making 12 golf trophies. Each trophy has a golf ballon it that has 300 dimples that have to be individually painted.If it takes Erin 3 seconds to completely paint one dimple and one second for every 3 dimples to dab her brush, how manyminutes will it take Erin to paint all of the dimples?18. The bottom cup in a 73 cm tall stack of nested identical cups is10 cm tall. If each additional cup adds 1.5 cm to the height ofthe stack, how many total cups are in the stack?19. Two interior angles of a convex pentagon are right angles andthe other three interior angles are congruent. In degrees, what is the measure of one of the three congruent interior angles? 20. If three people are selected at random from a group of sevenmen and three women, what is the probability that at least one woman is selected? Express your answer as a common fraction.21. Suzanne walks four miles every third day. What is the fewestnumber of miles she can walk in February?22. Mrs. Campbell had an 11:00 am appointment for a jobinterview. The company was 15 miles from her home. Sheaveraged 50 miles per hour for her trip and arrived 30 minutes early for her appointment. What time did she leave her house that morning? Express your answer to the nearest minute. 23. The sum of two integers is 75. One of the numbers is threemore than the other. What is the larger integer?12. ________________13. ________________14. ________________15. ________________16. ________________17. ________________18. ________________19. ________________20. ________________21. ________________22. ________________23. ________________(votes)(percent)(minutes)(cups)(degrees)(miles)(am)24. If 4 wands are equivalent to 6 rands and 24 rands are equivalent to 8 fands, how many wands are equivalent to 5 fands? 25. If two prime numbers are roots of the equation x 2 −12x + k = 0, what is the value of k ? 26. If a is 200% of b , what percent of 5a is 4b ?27. Rectangle WXYZ is drawn on ΔABC, suchthat point W lies on segment AB, point X lies on segment AC, and points Y andZ lie on segment BC, as shown. If m ∠BWZ = 26° and m ∠CXY = 64°, what is m ∠BAC, in degrees?28. If x is tripled and then increased by 7, the result is −8. What isthe value of x ? 29. Two-thirds of the people in a room are seated and one-quarter of the chairs are empty. If there are 6 empty chairs, how many people are in the room? 30. Two sides of a triangle have lengths 23 and 53. The third side has length n 2, where n is a positive integer. How many possible values are there for n ? 31. What is the volume, in cubic inches, of a rectangular box, whose faces have areas of 24 square inches, 16 square inches and 6 square inches? 32. Five plus 500% of 10 is the same as 110% of what number? 33. How many non-empty subsets of {−2, −1, 0, 1, 2} would have a sum of 0? 34. For how many positive integer values for k in the equation kx + 30 = 6k is the value of x also a positive integer? 35. Six small circles, each of radius 3 units, are tangent to a large circle as shown. Each small circle also is tangent to its two neighboring small circles. What is the diameter of the large circle in units?24. ________________25. ________________26. ________________27. ________________28. ________________29. ________________30. ________________31. ________________32. ________________33. ________________34. ________________35. ________________(wands)(percent)(degrees)(subsets)(units)(people)(values)(cu inches)(values)A BC WX Y Z26º64º47. A girl is half as old as her sister and is also two years younger than her brother. If the sum of the three children’s ages is 34 years, what is the product of their ages? 48. Given the set {4, 6, 8, 10, A, B} of six distinct integers, what is the product of A and B if the mean and median of the set are both 7 and the range is 6? 49. In 2004, 50 out of every 100 drivers at the National Trucking Company passed their driver’s license exam on their first try. In 2005, 62% of the drivers passed on their first attempt. What was the percent increase in the passing rate? 50. The measure of each exterior angle of a regular polygon is 30 degrees. What is the sum of the measures of the interior angles, in degrees? 51. What is the sum of all of the positive factors of 36?52. The seventh and tenth terms of a geometric sequence are 7 and 21, respectively. What is the 13th term of this progression? 53. If x is doubled, increased by 3, and then divided by 5, the result is 11. What is the value of x ? 54. There are two rectangles with integral dimensions whose areas, in square units, are numerically equal to their perimeters, in units. What is the positive difference, in units, between the perimeters of these two rectangles? 55. How many positive three-digit integers with a 5 in the units place are divisible by 15? 56. What is the value of x in the equation1616 + 1616 + 1616 + 1616 = 2x ?57. A particular novel contains 40,000 words. If the authorcompleted the novel in 80 hours, on average how many words per hour did she write? 58. What is the perimeter, in cm, of quadrilateral ABCD if AB BC ⊥, DC BC ⊥, AB = 9 cm, DC = 4 cm, and BC = 12 cm?59. What is the largest positive integer with only even digits that isless than 10,000 and is a multiple of 9?47. ________________48. ________________49. ________________50. ________________51. ________________52. ________________53. ________________54. ________________55. ________________56. ________________57. ________________58. ________________59. ________________(percent)(degrees)(years 3)ABCD(units)(three-digitintegers)(words perhour)(cm)60. If a pyramid has 14 edges, how many vertices does it have?61. The pattern, shown, is folded along thedashed lines to make a right triangular prism.What is the volume, in cubic units, of the triangular prism? 62. If m and n are positive integers and 3m + 9m + 27n + 81n = 204, what is the value of the product mn ? 63. What is the area, in square feet, of an isosceles triangle whose vertex angle is 120° and whose base is 20 feet long? Express your answer as a common fraction in simplest radical form. 64. What is 2000% – 200% + 20% – 2%? Express your answer as a decimal to the nearest hundredth. 65. Ten more than five times x equals five less than ten times x . What is the value of x ? 66. Triangle ABC has sides of 6 units, 8 units and 10 units. The width of a rectangle, whose area is equal to the area of the triangle, is 4 units. What is the perimeter of this rectangle, in units? 67. What is the probability that at least two of the faces matchwhen you roll three fair six-sided dice? Express your answer as a common fraction. 68. The sum of three consecutive even integers is 66. What is the smallest of the three integers? 69. The sum of two numbers is 30. The difference of twice the larger number and three times the smaller number is 5. What is the positive difference between the two numbers? 70. At a driver education class, the students were told to put their hands on the steering wheel on “3” and “9,” just as on the face of a clock. If, instead, the students placed their hands on the “2” and the “10,” by how many degrees would the angle between the hands (with the center of the wheel as the vertex of the angle) decrease? 71. “Buy 3, get 2 free” is equivalent to purchasing five at a discount of what percent?60. ________________61. ________________62. ________________63. ________________64. ________________65. ________________66. ________________67. ________________68. ________________69. ________________70. ________________71. ________________(vertices)(square feet)(units)(degrees)(cubic units)(percent)151213121372. The price of a particular item is reduced by successivediscounts of 30%, 40% and 50%. What is the selling price of the item after the three successive discounts, if it originally cost $200?73. When the positive difference of the reciprocals of 8 and 4 isadded to the reciprocal of 10, what is the result? Express your answer as a common fraction.74. The speed of light in a vacuum is 186,000 miles per second. Ifthe speed of light in glass is 124,000 miles per second, what is the ratio of the speed of light in glass to the speed of light in a vacuum? Express your answer as a common fraction.75. How many distinct, positive factors does 1100 have?76. What is the least common multiple of 14, 20 and 35?77. One more than 11 times a certain prime p is another prime q.What is the value of q?78. The sum of two numbers is 12 and their difference is 20. Whatis the smaller number?79. What is the sum of the first ten positive multiples of 13?80. Mary’s age is twice her brother’s age and half her sister’s age.How old is Mary, in years, if the sum of the three ages is 28years? 72. ________________73. ________________74. ________________75. ________________76. ________________77. ________________78. ________________79. ________________80. ________________(dollars)(distinctfactors)(years)。

Problem 1Each morning of her five-day workweek, Jane bought either a 50-cent muffin or a 75-cent bagel. Her total cost for the week was a whole number of dollars, How many bagels did she buy?SolutionProblem 2Which of the following is equal to ?SolutionProblem 3Paula the painter had just enough paint for identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for rooms. How many cans of paint did she use for the rooms?SolutionProblem 4A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths and meters. What fraction of the yard is occupied by the flower beds?SolutionProblem 5Twenty percent less than 60 is one-third more than what number?SolutionProblem 6Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?SolutionProblem 7By inserting parentheses, it is possible to give the expressionseveral values. How many different values can be obtained?SolutionProblem 8In a certain year the price of gasoline rose by during January, fell byduring February, rose by during March, and fell by during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is ?SolutionProblem 9Segment and intersect at , as shown, , and . What is the degree measure of ?SolutionProblem 10A flagpole is originally meters tall. A hurricane snaps the flagpole at a point meters above the ground so that the upper part, still attached to the stump, touches the ground meter away from the base. What is ?SolutionProblem 11How many -digit palindromes (numbers that read the same backward as forward) can be formed using the digits , , , , , , ?SolutionProblem 12Distinct points , , , and lie on a line, with . Points and lie on a second line, parallel to the first, with . A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?SolutionProblem 13As shown below, convex pentagon has sides , , , , and . The pentagon is originally positioned in the plane with vertex at the origin and vertex on the positive -axis. The pentagon is then rolled clockwise to the right along the -axis. Which side will touch the point on the -axis?SolutionProblem 14On Monday, Millie puts a quart of seeds, of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?SolutionProblem 15When a bucket is two-thirds full of water, the bucket and water weigh kilograms. When the bucket is one-half full of water the total weight is kilograms. In terms of and , what is the total weight in kilograms when the bucket is full of water?SolutionProblem 16Points and lie on a circle centered at , each of and are tangent to the circle, and is equilateral. The circle intersects at . What is?SolutionProblem 17Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from to , divides the entire region into two regions of equal area. What is ?SolutionProblem 18Rectangle has and . Point is the midpoint of diagonal , and is on with . What is the area of ?SolutionProblem 19A particular -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a , it mistakenly displays a . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?SolutionProblem 20Triangle has a right angle at , , and . The bisector of meets at . What is ?SolutionProblem 21What is the remainder when is divided by 8?SolutionProblem 22A cubical cake with edge length inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where is the midpoint of a top edge. The piece whose top is triangle contains cubic inches of cake and square inches of icing. What is ?SolutionProblem 23Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?SolutionProblem 24The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with trapezoids, let be the angle measure in degrees of the larger interior angle of the trapezoid. What is ?SolutionProblem 25Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?。

2008年 第9届 美国AMC10(2008年2月 日 时间75分钟)1. 某面包店老板在上午8：30启动甜甜圈制作机，在上午11：10这机器已完成当天三分之一的工作。

请问该机器何时可以完成当天全部的工作？(A) 下午1：50 (B) 下午3：00 (C) 下午3：30 (D) 下午4：30 (E) 下午5：50 。

2. 在长方形内昼一个正方形。

长方形的宽与正方形的边长比为2：1，长方形的长与宽之比为2：1。

请问正方形面积占长方形面积的百分之多少？ (A) 12.5 (B) 25 (C) 50 (D) 75 (E) 87.5 。

3. 对于正整数n ，以<n >表示除了n 本身之外其所有正因子的和。

例如：<4>=1+2=3，而<12>=1+2+3+4+6=16。

请问<<<6>>>=？ (A) 6 (B) 12 (C) 24 (D) 32 (E) 36 。

4. 若10根香蕉价钱的32与8个橘子的价钱相同，请问多少个橘子的价钱会与5根香蕉价钱的21相同？ (A) 2 (B) 25 (C) 3 (D)27 (E) 4 。

5. 请问分式48⨯812⨯…⨯nn 444+⨯…⨯20042008的乘积是多少？(A) 251 (B) 502 (C) 1004 (D) 2008 (E) 4016 。

6. 某选手参加游泳、单车、赛跑三项距离都相等的三项运动竞赛，这名选手游泳的速率为每小时3公里，骑单车的速率为每小时20公里，跑步的速率为每小时10公里。

请问下列何者最接近这位选手整个赛程的平均速率(公里/小时)？ (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 。

7. 请问分式22005220072200622008)3()3()3()3(--可化简为何？ (A) 1 (B)49 (C) 3 (D)29 (E) 9 。

8. 小华比较某种新计算机在两家商店的价格。

2000到2012年A M C10美国数学竞赛P 0 A 0B 0 C0 D 0 全美中学数学分级能力测验(AMC 10)2000年 第01届 美国AMC10 (2000年2月 日 时间75分钟)1. 国际数学奥林匹亚将于2001年在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ?M ?O =2001，则试问I ?M ?O 之最大值为 。

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

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

(A) 40 (B) 50 (C) 55 (D) 60 (E) 754. Candra 每月要付给网络公司固定的月租费及上网的拨接费，已知她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是 元。

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

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

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

1985 年美国大学生数学建模竞赛MCM 试题1985年MCM:动物种群选择合适的鱼类和哺乳动物数据准确模型。

模型动物的自然表达人口水平与环境相互作用的不同群体的环境的重要参数,然后调整账户获取表单模型符合实际的动物提取的方法。

包括任何食物或限制以外的空间限制,得到数据的支持。

考虑所涉及的各种数量的价值,收获数量和人口规模本身,为了设计一个数字量代表的整体价值收获。

找到一个收集政策的人口规模和时间优化的价值收获在很长一段时间。

检查政策优化价值在现实的环境条件。

1985年MCM B:战略储备管理钴、不产生在美国,许多行业至关重要。

(国防占17%的钴生产。

1979年)钴大部分来自非洲中部,一个政治上不稳定的地区。

1946年的战略和关键材料储备法案需要钴储备,将美国政府通过一项为期三年的战争。

建立了库存在1950年代,出售大部分在1970年代初,然后决定在1970年代末建立起来,与8540万磅。

大约一半的库存目标的储备已经在1982年收购了。

建立一个数学模型来管理储备的战略金属钴。

你需要考虑这样的问题:库存应该有多大?以什么速度应该被收购?一个合理的代价是什么金属?你也要考虑这样的问题:什么时候库存应该画下来吗?以什么速度应该是画下来吗?在金属价格是合理出售什么?它应该如何分配?有用的信息在钴政府计划在2500万年需要2500万磅的钴。

美国大约有1亿磅的钴矿床。

生产变得经济可行当价格达到22美元/磅(如发生在1981年)。

要花四年滚动操作,和thsn六百万英镑每年可以生产。

1980年,120万磅的钴回收,总消费的7%。

1986 年美国大学生数学建模竞赛MCM 试题1986年MCM A:水文数据下表给出了Z的水深度尺表面点的直角坐标X,Y在码(14数据点表省略)。

深度测量在退潮。

你的船有一个五英尺的草案。

你应该避免什么地区内的矩形(75200)X(-50、150)?1986年MCM B:Emergency-Facilities位置迄今为止,力拓的乡牧场没有自己的应急设施。

环岛交通的优化设计Optimal Design for Traffic CircleAbstract欧仁艾纳尔（Ez-gene Herlerd ）受到19世纪艺术形式的影响首创了“环岛式交通枢纽”的道路交叉口的概念。

时至今日，由此产生的环岛式交通在世界各地广泛存在。

一些城市主干道上现有的环形交叉口由于通行能力不足，经常出现拥挤、混乱及堵塞的现象，往往是各向车辆争相进交叉口，却很难顺畅驶出。

面对这种状况，我们以六车道入口环岛为例建立了三个模型：1、通过对六车道环岛模型的分析，考虑不加任何控制设施建立模型，并推导出环岛的交通能力表达式（考虑到交织段长度影响和车辆分布不均匀的影响及非机动车干扰）：22()3600132()(11/6)(11/6)(11/6)230A A Z i Q Q Q A l Q Q p p t p l βββ''''-∆•=•=•=•-∆•••---+ 以某一环岛为例，代入上式求出此种情况下的通行能力。

2、建立指示牌控制通行能力的计算与信号灯控制通行能力计算的模型，并以上面提到的环岛为例计算出两种模式下通行能力，同时与上面模型比较，从而对交通工程师提出指导性建议。

3、信号灯控制模型, 根据交通状况的实际需求,以延误最小、停车最少和通行能力最大作为目标函数,利用可随交通需求实时变化的加权系数把这3个目标统一为单目标函数, 建立交叉口信号配时非线性优化模型如下:224(25/)1/321[(1.0)]min (,)2(2)0.65()2(1.0)2(1)0.9(1.0/)/(1.0)2(/3600)(/)2i x c i i i i i i i i i i c x c y c Z x c Y y y q y q c x c y c Y x c s +=-=-⋅+-+--⋅⋅---⋅⋅⋅∑ 在引入算例的情况下，将算例所提供的数据代入优化得到的模型，采用基于精英蚂蚁寻优策略进行求解。

2013-2014年度美国“数学大联盟杯赛”(中国赛区)初赛答案(三、四年级)一、选择题1. A.Since 0 is a factor, 2 × 0 × 1 × 4 = 0.A) 0B) 7C) 8D) 20142. B.If 2 years ago I was 3 years old, I am now 3 + 2 = 5.A) 4B) 5C) 6D) 233. D.8 + (60 ÷ 4) = 8 + (15) = 23.A) 15B) 17C) 22D) 234. B.(1 + 7) + (2 + 6) + (3 + 5) = 3 × 8 = 24 = 4 + 20.A) 4B) 20C) 24D) 285. C.The prime numbers less than 10 are 2, 3, 5, and 7.A) 2B) 3C) 4D) 56. B.Caleb the dog dreams he has 12 dozen bones. Since 12 dozen = 12 × 12 = 144, there are 144 ÷ 2 = 72 pairs. Caleb will have to dig 72 holes.A) 24B) 72C) 144D) 2887. C.From 9:45 PM to 10:45 PM is 60 mins. From 10:45 PM to 11 PM is 15 mins. From 11 PM to 11:10 PM is 10 mins. That’s (60 + 15 + 10) mins.A) 65B) 75C) 85D) 958. D.From January 1st to January 31st, there are 16 odd-numbered dates. From February 1st to February 21st, there are 11 odd-numbered dates. That’s 27 × $2 = $54.A) $48B) $50C) $52D) $549. C.9 × 9 + 9 × 8 + 9 × 7 + 9 × 6 = 9 × (9 + 8 + 7 + 6).A) 20B) 24C) 30D) 3610.D.Manny weighs three times as much as Murray. Manny also weighs 8000 kg more than Murray, so 8000 kg is twice Murray’s weight. Thus Murray weighs 4000 kg and Manny weighs 12 000 kg.D) 12 00011.B.I have twice as many shirts as hats, and four times as many hats as scarves. If I have 24 shirts, I have 24÷ 2 = 12 hats and 12 ÷ 4 = 3 scarves.A) 2B) 3C) 6D) 1212.C.My coins have a total value of $6.20. If I have 1 of each coin, I have (1 + 5 + 10 + 25)¢ = 41¢. Subtract 41¢ from $6.20 repeatedly until there is 5¢ left. After 15 subtractions, there is 5¢ left. I have 15 + 5 or20 pennies.A) 10B) 15C) 20D) 2513.D.The diagrams demonstrate choices A, B, and C.A) 14 kmB) 10 kmC) 8 kmD) 1 km14.C.(2014 −1014) + (3014 − 2014) = 1000 + 1000 = 2000.A) 0B) 1000C) 2000D) 201415.A.10 + (9 ×8) − (7 × 6) = 10 + 72 − 42 = 40.A) 40B) 110The prime factorization of 72 is 2 × 2 × 2 × 3 × 3. The largest prime is 3.A) 3B) 7C) 36D) 7217.D.6 × 4 = 24 = 96 ÷ 4.A) 6B) 12C) 24D) 9618.C.If 6 cans contain 96 teaspoons of sugar, 1 can contains 96 ÷ 6 = 16 teaspoons of sugar. Thus 15 cans contain 16 × 15 = 240 teaspoons of sugar.A) 192B) 208C) 240D) 28819.C.The largest possible such sum is 98 + 99 = 197.A) 21B) 99C) 197D) 19820.B.Ann sent Wilson hearts with odd numbers with odd tens digits. The number on each heart he received must be two digits with both digits odd. There are 5 possible tens digits and 5 possible ones digits.That’s a total of 5 × 5 = 25 hearts.A) 23B) 25C) 30D) 4521.B.Since Rich ate his favorite sandwich 8 days ago, today is the 9th day of the month. Since the shortest month has 28 days, it is at least 28 − 9 = 19 days until the last day of the month. He must wait 1 more day.A) 1922.D.The factors of 49 are 1, 7, and 49. Since 49 has 3 factors, it has a prime number of factors.A) 6B) 12C) 36D) 4923.D.Dividing a certain two-digit number by 10 leaves a remainder of 9, so it is 19, 29, 39, 49, 59, 69, 79, 89, or 99. The only number listed with remainder 8 when divided by 9 is 89, so the number is 89 and 8 + 9 = 17.A) 7B) 9C) 13D) 1724.A.The whole numbers less than 1000 that can be written as such a product are 0 × 1 × 2, 1 × 2 × 3, 2 × 3 ×4, 3 × 4 × 5, 4 × 5 × 6, 5 × 6 × 7, 6 × 7 × 8, 7 × 8 × 9, 8 × 9 × 10, and 9 × 10 ×11. In all, that’s 10.A) 10B) 11C) 15D) 2125.B.The only such numbers are 5432, 5431, 5430, 5421, 5420, 5410, 5321, 5320, 5310, and 5210. In all, there are 10 such numbers.A) 3B) 10C) 69D) 12026.C.2014 × 400 = 805 600; the hundreds digit is 6.A) 0B) 5C) 6D) 827.B.Greta was 110 cm tall 2 years ago, when she was 10 cm taller than her brother. Her brother was 100 cmB) 130C) 140D) 15028.B.The number 789 678 567 456 is added to the number 987 876 765 654. Since we carry a 1 when adding the left-most digits, the sum has 12 + 1 digits.A) 12B) 13C) 24D) 2529.D.We must find which number among the choices is two more than a multiple of 5. Divide each choice by5 (or recogniz e that any number that ends in “2”or “7” is 2 more than a multiple of 5).A) 4351B) 5215C) 5616D) 646230.C.Of every 11 people, there are 2 adults and 9 children. Since 99 ÷ 11 = 9, there are 9 groups of 11 people.Of these, 9 × 2 = 18 are adults.A) 9B) 11C) 18D) 22二、填空题31.5.32.22.33.4.34.1.35.617.36.21.37.499.38.765.39.69.40.10.。

2010 年美国大学生数学建模竞赛MCM、ICM 试题2010 MCM A: The Sweet Spot Explain the “sweet spot”on a baseball bat. Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding. Some players bel ieve that “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major Leagu e Baseball prohibits “corking”? Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats? 2010 MCM B: Criminology In 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to find a “center of mass” of the locatio ns of the attacks. In the end, the suspect happened to live in the same town predicted by this technique. Since that time, a number of more sophisticated techniques have been developed to determine the “geographical profile” of a suspected serial criminal based on the locations of the crimes. Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of at least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful prediction for law enforcement officers. The prediction should provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, you must provide specific details about how you incorporate the extra information. Your method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings. In addition to the required one-page summary, your report should include an additional two-page executive summary. The executive summary should provide a broad overview of the potential issues. It should provide an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool. The executive summary will be read by a chief of police and should include technical details appropriate to the intended audience. 2010 ICM: The Great Pacific Ocean Garbage Patch。