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2010年北美数学建模比赛中英文题目(MCM)2010 MCM题目A题：棒球棒上的最佳击球点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 believe 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 League Baseball prohibit s “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?中文翻译：解释棒球棒上的“最佳击球点”。
1.What is the value of下方表达式的值是多少？(A) 0 (B) 1 (C) 2 (D) 3 (E) 42.What is the hundreds digit of (20! − 15!) ? (20! − 15!)的百位上的数字是多少？ (A) 0 (B) 1 (C) 2 (D) 4(E) 53.Ana and Bonita were born on the same date in different years, n years apart. Last year Ana was 5 times as old as Bonita. This year Ana’s age is the square of Bonita’s age. What is n ?Ana 和Bonita 出生在不同年份的同一天, 相隔n 年。
去年Ana 的年龄是Bonita 年龄的5倍。
今年Ana 的年龄是Bonita 年龄的平方。
问n 的值是多少？(A) 3 (B) 5 (C) 9 (D) 12 (E) 154.A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn?一个盒子中有28个红球，20个绿球，19个黄球，13个蓝球，11个白球和9个黑球。
为了保证至少取出15个单一颜色的球, 在不允许放回重取的情况下，须从盒子里至少取出多少个球？(A) 75(B) 76(C) 79(D) 84(E) 915.What is the greatest number of consecutive integers whose sum is 45 ? 最多可以有多少个连续整数的和是45?(A) 9 (B) 25 (C) 45 (D) 90 (E) 1206. For how many of the following types of quadrilaterals does there exist a point in the plane of thequadrilateral that is equidistant from all four vertices of the quadrilateral?2019 AMC 10A Problems • a square• a rectangle that is not a square• a rhombus that is not a square• a parallelogram that is not a rectangle or a rhombus•an isosceles trapezoid that is not a parallelogram 在以下类型的四边形中，有多少种在平面上存在一个与其所有四个顶点距离都相等的点？• 正方形• 不是正方形的长方形• 不是正方形的菱形• 不是长方形或菱形的平行四边形• 不是平行四边形的等腰梯形(A) 1(B) 2 (C) 3 (D) 4 (E) 57. Two lines with slopes and intersect at . What is the area of the triangle enclosed by these two lines and the line两条斜率分别为1/2和2的直线相交于(2, 2)。
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）。
2000到2012年AMC10美国数学竞赛0 0P 0 A 0 B 0 C 0D 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) △P AB 之周长 (c) △P AB 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 。
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）。
USA 2010 A1. 小华的书架顶上的5本书的厚度分别是：, , , , 公分，请问这些书的平均厚度是多少公分？（A）1 (B)2 (C)3 (D)4 (E)52. 四个相同的小正方形和一个长方形形成一个大正方形，如下所示。
长方形中长是宽的多少倍？3. 小明有97颗弹珠和小钟有11颗弹子。
然后小明把他的一些弹珠给了小钟，所以小明有小钟2倍的弹珠。
小明给了小钟多少弹珠？（A）3 (B)13 (C)18 (D)25 (E)294.某本书转录成有声书需要报读412分钟，现在要将它储存在光盘片里。
已知每片光盘最多可以储存报读时间为56分钟，假设用最少片的光盘，使得每片光盘所储存报读的时间都相同。
试问每片光盘应该储存报读的时间是多少分钟？（A）50.2 (B)51.5 (C)52.4 (D)53.8 (E)55.25.若圆周长为之圆的面积为，那么K数值是多少？（A）6 (B)12 (C)24 (D)36 (E)1446. 对于正数X,Y，定义运算如下：。
试问是多少？7.小美每天一招固定路线跑步，她首先朝正北跑了1公里，然后她跑东北方向1公里，然后东南方向1公里。
她最后按照直线方向跑回原出发点。
她最后一段跑了多少公里？8.小明一天工作2小时，每小时工资0. 50美元乘以他工作时的年龄。
他在六个月的期间，一共工作了50天，共赚630美元。
试问他在六个月工作结束时年龄是多少岁？9.如果一个数由左至右念与由右至左念都一样，我们称它为回文数，例如83438，已知X与 X+32分别为三位数与四位数的回文数，试问X各位数字的总和是多少？10.小华的生日是5月27日，在闰年2008年的生日是星期二。
试问未来他的生日在星期六的最近一年是哪一年？11. 设不等式解的范围之长度为.试问是多少?12.小罗按照比例缩小尺寸制作他所居住城镇的模型。
城镇的水塔高40公尺，顶部是一个球形储水槽，拥有100000公升的水。
若小罗模型水塔的储水槽可存0.1升。
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。
2000到20XX年AMC10美国数学竞赛0 0P 0 A 0 B 0 C 0D 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 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) △P AB 之周长 (c) △P AB 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 。
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卷时间：2021.03.03 创作：欧阳学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)(B)(C)(D)(E)4. 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 ?(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) (C) (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) (B) (C)(D) (E)8. Last summer 30% of the birds living on TownLake 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 allpositive 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)(B)(C)(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 morethan 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 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 th e numerical value of the circle’s circumference?(A)(B)(C)(D)(E)15. 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)(D)(E)17. 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 of tangency. 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)(B)(C)(D)(E)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)(B)(C)(D)(E)21. 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)(B)(C)(D)(E)22. Each vertex of convex pentagon ABCDE is to be assigneda 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 consecutivegrope 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)(B)(C)(D)(E)25. Let R be a square region and 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）。
