2010年AMC 10B竞赛真题及答案(英文版)
2010年2012 美国AMC10
2010AMC 10 答案： 1. (D) 2. (B) 3. (D) 4. (B) 5. (E) 6. (C) 7. (C) 8. (D) 9. (E) 10. (E) 11. (D) 12. (C) 13. (A) 14. (C) 15. (D) 16. (B) 17. (A) 18. (B) 19. (E) 20. (D) 21. (A) 22. (A) 23. (A) 24. (A) 25. (B)
25. 小甄由正整数 n 开始，造一个数列。每一项是由前一项减去最大可能的平方数，此平方数 小于或等于前一项，直到等于 0 停止。例如，小甄从 n=55 开始，则此数列有 5 项如下： 55 2 557 =6 622=2 212=1 112=0 设小甄从 N 开始所造的数列有 8 项。试问最小的 N 其个位数字为何？ (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 。
2010 年 第 11 届 美国 AMC10 (2010 年 2 月
1. 小华书架上放了五本书，各书的厚度分别为 6， 厚度是多少公分？ (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 。 2. 四个一样大小的正方形与一个长方形拼成一个大正方形，如右图所示。 试问长方形中长是宽的多少倍？ (A)
1 的最小 n 是多少？ (A) 45 (B) 63 (C) 64 (D) 201 (E) 1005 。 2010
24. 设 90！乘开后最后不为 0 的两个数字所形成的二位数为 n。试问 n 为下列哪一个数？ (注：k！=k(k1)(k2)…21) (A) 12 (B) 32 (C) 48 (D) 52 (E) 68 。
2 (C)
3 (D) 2 (E) 2 2 。
8. 东尼一天工作 2 小时，每小时的工资是美金 0.5 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，总共赚得美金 630 元。试问他在六个月工作结束时的年龄为几岁？ (A) 9 (B) 11 (C) 12 (D) 13 (E) 14 。 9. 如果一个数由左至右念与由右至左念都相同，我们就称它为回文数，例如 83438 就是一个回 文数。已知 x 与 x32 分别为三位数与四位数的回文数。试问 x 各位数字的和为多少？ (A) 20 (B) 21 (C) 22 (D) 23 (E) 24 。 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？ (A) 2011 (B) 2012 (C) 2013 (D) 2015 (E) 2017 。 11. 设不等式 a2x3b 解的范围之长度为 10。试问 ba=？ (A) 6 (B) 10 (C) 15 (D) 20 (E) 30 。 12. 小罗要按比例缩小尺寸制作他所住城镇的模型。他所住城镇的水塔高 40 公尺，顶端是一个 球形的储水槽，可储存 100,000 公升的水。若小罗模型水塔的储水槽可储存 0.1 公升的水， 则小罗模型水塔的高应是多少公尺？ (A) 0.04 (B)
2010 AMC 10A 试题及答案解析
2010 AMC 10A problems and solutions.The test was held on February 8, 2010. The first link contains the full set of test problems. The rest contain each individual problem and its solution.Problem 1Mary’s top book shelf holds five books with the follow ing widths, incentimeters: , , , , and . What is the average book width, in centimeters?SolutionTo find the average, we add up the widths , , , , and , to get a total sum of . Since there are books, the average book width isThe answer is .Problem 2Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?SolutionLet the length of the small square be , intuitively, the length of the big square is . It can be seen that the width of the rectangle is .Thus, the length of the rectangle is times large as the width. The answer is .Problem 3Tyrone had marbles and Eric had marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?SolutionLet be the number of marbles Tyrone gave to Eric. Then,. Solving for yields and . The answer is .Problem 4A book that is to be recorded onto compact discs takes minutes to read aloud. Each disc can hold up to minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?SolutionAssuming that there were fractions of compact discs, it would take CDs to have equal reading time. However, since the number of discs can only be a whole number, there are at least 8 CDs, in which case it would have minutes on each of the 8 discs. The answer is .Problem 5The area of a circle whose circumference is is . What is the value of ?SolutionIf the circumference of a circle is , the radius would be . Since the area of a circle is , the area is . The answer is . Problem 6For positive numbers and the operation is defined asWhat is ?Solution. Then, is The answer isProblem 7Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?SolutionCrystal first runs North for one mile. Changing directions, she runs Northeast for another mile. The angle difference between North and Northeast is 45 degrees. She then switches directions to Southeast, meaning a 90 degree angle change. The distance now from travelling North for one mile, and her current destination is miles, because it is the hypotenuse of a 45-45-90 triangle with side length one (mile). Therefore, Crystal's distance from her starting position, x, is equal to , which is equal to . The answer isTony works hours a day and is paid $per hour for each full year of his age. During a six month period Tony worked days and earned $. How old was Tony at the end of the six month period?SolutionTony worked hours a day and is paid dollars per hour for each full year of his age. This basically says that he gets a dollar for each year of his age. So if he is years old, he gets dollars a day. We also know that he worked days and earned dollars. If he was years old at the beginning of his working period, he would have earned dollars. If he was years old at the beginning of his working period, he would have earned dollars. Because he earned dollars, we know that he was for some period of time, but not the whole time, because then the money earned would be greater than or equal to . This is why he was when he began, but turned sometime in the middle and earned dollars in total. So the answer is .The answer is . We could find out for how long he was and . . Then isand we know that he was for days, and for days. Thus, the answer is .Problem 9A palindrome, such as , is a number that remains the same when its digits are reversed. The numbers and are three-digit and four-digit palindromes, respectively. What is the sum of the digits of ?Solutionis at most , so is at most . The minimum value ofis . However, the only palindrome between and is , which means that must be .It follows that is , so the sum of the digits is .Marvin had a birthday on Tuesday, May 27 in the leap year . In what year will his birthday next fall on a Saturday?Solution(E) 2017There are 365 days in a non-leap year. There are 7 days in a week. Since 365 = 52 * 7 + 1 (or 365 is congruent to 1 mod 7), the same date (after February) moves "forward" one day in the subsequent year, if that year is not a leap year.For example: 5/27/08 Tue 5/27/09 WedHowever, a leap year has 366 days, and 366 = 52 * 7 + 2. So the same date (after February) moves "forward" two days in the subsequent year, if that year is a leap year.For example: 5/27/11 Fri 5/27/12 SunYou can keep count forward to find that the first time this date falls on a Saturday is in 2017:5/27/13 Mon 5/27/14 Tue 5/27/15 Wed 5/27/16 Fri 5/27/17 Sat Problem 11The length of the interval of solutions of the inequality is . What is ?SolutionSince we are given the range of the solutions, we must re-write the inequalities so that we have in terms of and .Subtract from all of the quantities:Divide all of the quantities by .Since we have the range of the solutions, we can make them equal to .Multiply both sides by 2.Re-write without using parentheses.Simplify.We need to find for the problem, so the answer isProblem 12Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?SolutionThe water tower holds times more water than Logan's miniature. Therefore, Logan should make his towertimes shorter than the actual tower. This ismeters high, or choice .Problem 13Angelina drove at an average rate of kph and then stopped minutes for gas. After the stop, she drove at an average rate of kph. Altogether she drove km in a total trip time of hours including the stop. Which equation could be used to solve for the time in hours that she drove before her stop?SolutionThe answer is （）because she drove at kmh for hours (the amount of time before the stop), and 100 kmh for because she wasn't driving for minutes, or hours. Multiplying by gives the total distance, which is kms. Therefore, the answer isProblem 14Triangle has . Let and be on and , respectively, such that . Let be the intersection of segments and , and suppose that is equilateral. What is ?SolutionLet .Since ,Problem 15In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.Brian: "Mike and I are different species."Chris: "LeRoy is a frog."LeRoy: "Chris is a frog."Mike: "Of the four of us, at least two are toads."How many of these amphibians are frogs?SolutionSolution 1We can begin by first looking at Chris and LeRoy.Suppose Chris and LeRoy are the same species. If Chris is a toad, then what he says is true, so LeRoy is a frog. However, if LeRoy is a frog, then he is lying, but clearly Chris is not a frog, and we have a contradiction. The same applies if Chris is a frog.Clearly, Chris and LeRoy are different species, and so we have at least frog out of the two of them.Now suppose Mike is a toad. Then what he says is true because we already have toads. However, if Brian is a frog, then he is lying, yet his statement is true, a contradiction. If Brian is a toad, then what he says is true, but once again it conflicts with his statement, resulting in contradiction.Therefore, Mike must be a frog. His statement must be false, which means that there is at most toad. Since either Chris or LeRoy is already a toad, Brain must be a frog. We can also verify that his statement is indeed false.Both Mike and Brian are frogs, and one of either Chris or LeRoy is a frog, so we have frogs total.Solution 2Start with Brian. If he is a toad, he tells the truth, hence Mike is a frog. If Brian is a frog, he lies, hence Mike is a frog, too. Thus Mike must be a frog.As Mike is a frog, his statement is false, hence there is at most one toad.As there is at most one toad, at least one of Chris and LeRoy is a frog. But then the other one tells the truth, and therefore is a toad. Hence we must have one toad and three frogs.Problem 16Nondegenerate has integer side lengths, is an angle bisector, , and . What is the smallest possible value of the perimeter?SolutionBy the Angle Bisector Theorem, we know that . If we use the lowest possible integer values for AB and BC (the measures of AD and DC, respectively), then , contradicting the Triangle Inequality. If we use the next lowest values (and ), the Triangle Inequality is satisfied. Therefore, our answer is , or choice .Problem 17A solid cube has side length inches. A -inch by -inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?SolutionSolution 1Imagine making the cuts one at a time. The first cut removes a box . The second cut removes two boxes, each of dimensions, and the third cut does the same as the second cut, on the last two faces. Hence the total volume of all cuts is .Therefore the volume of the rest of the cube is.Solution 2We can use Principle of Inclusion-Exclusion to find the final volume of the cube.There are 3 "cuts" through the cube that go from one end to the other. Each of these "cuts" has cubic inches. However, we can not just sum their volumes, as the central cube is included in each of these three cuts. To get the correct result, we can take the sum of the volumes of the three cuts, and subtract the volume of the central cube twice.Hence the total volume of the cuts is.Therefore the volume of the rest of the cube is.Solution 3We can visualize the final figure and see a cubic frame. We can find the volume of the figure by adding up the volumes of the edges and corners.Each edge can be seen as a box, and each corner can be seen as a box..Problem 18Bernardo randomly picks 3 distinct numbers from the setand arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?SolutionWe can solve this by breaking the problem down into cases and adding up the probabilities.Case : Bernardo picks . If Bernardo picks a then it is guaranteed that his number will be larger than Silvia's. The probability that he will pick a is .Case : Bernardo does not pick . Since the chance of Bernardo picking is , the probability of not picking is .If Bernardo does not pick 9, then he can pick any number from to . Since Bernardo is picking from the same set of numbers as Silvia, the probability that Bernardo's number is larger is equal to the probability that Silvia's number is larger.Ignoring the for now, the probability that they will pick the same number is the number of ways to pick Bernardo's 3 numbers divided by the number of ways to pick any 3 numbers.We get this probability to beProbability of Bernardo's number being greater isFactoring the fact that Bernardo could've picked a but didn't:Adding up the two cases we getProblem 19Equiangular hexagon has side lengthsand . The area of is of the area of the hexagon. What is the sum of all possible values of ?SolutionSolution 1It is clear that is an equilateral triangle. From the Law of Cosines, we get that . Therefore, the area of is .If we extend , and so that and meet at , and meet at , and and meet at , we find that hexagon is formed by taking equilateral triangle of side length and removing three equilateral triangles, , and , of side length . The area of is therefore.Based on the initial conditions,Simplifying this gives us . By Vieta's Formulas we know that the sum of the possible value of is .Solution 2As above, we find that the area of is .We also find by the sine triangle area formula that, and thusThis simplifies to.Problem 20A fly trapped inside a cubical box with side length meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?SolutionThe distance of an interior diagonal in this cube is and the distance of a diagonal on one of the square faces is . It would not make sense if the fly traveled an interior diagonal twice in a row, as it would return to the point it just came from, so at most the final sum can only have 4 as the coefficient of . The other 4 paths taken can be across a diagonal on one of the faces, so the maximum distance traveled is.Problem 21The polynomial has three positive integer zeros. What is the smallest possible value of ?SolutionBy Vieta's Formulas, we know that is the sum of the three roots of the polynomial . Also, 2010 factors into. But, since there are only three roots to the polynomial, two of the four prime factors must be multiplied so that we are left with three roots. To minimize , and should be multiplied, which means will be and the answer is .Problem 22Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point insidethe circle. How many triangles with all three vertices in the interior of the circle are created?SolutionTo choose a chord, we know that two points must be chosen. This implies that for three chords to create a triangle and not intersect at a single point, six points need to be chosen. Therefore, the answer is which is equivalent to 28,Problem 23Each of 2010 boxes in a line contains a single red marble, and for , the box in the position also contains white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let be the probability that Isabella stops afterdrawing exactly marbles. What is the smallest value of for which ?SolutionThe probability of drawing a white marble from box is . Theprobability of drawing a red marble from box is .The probability of drawing a red marble at box is thereforeIt is then easy to see that the lowest integer value of that satisfies the inequality is .Problem 24The number obtained from the last two nonzero digits of is equal to . What is ?SolutionWe will use the fact that for any integer ,First, we find that the number of factors of in is equal to. Let . The we want is therefore the last two digits of , or . Since there is clearly an excess of factors of 2, we know that , so it remains to find .If we divide by by taking out all the factors of in , we canwrite as where where every multiple of 5 is replaced by the number with all its factors of 5 removed. Specifically, every number in the form is replaced by , and every number in the form is replaced by .The number can be grouped as follows:Using the identity at the beginning of the solution, we can reducetoUsing the fact that (or simply the fact that if you have your powers of 2 memorized), we can deduce that . Therefore.Finally, combining with the fact that yields.Problem 25Jim starts with a positive integer and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with , then his sequence contains numbers:Let be the smallest number for which Jim’s sequence has numbers. What is the units digit of ?SolutionWe can find the answer by working backwards. We begin with on the bottom row, then the goes to the right of the equal's sign in the row above. We find the smallest value for whichand , which is .We repeat the same procedure except with for the next row and for the row after that. However, at the fourth row, wesee that solving yields , in which case it would be incorrect since is not the greatest perfect square less than or equal to . So we make it a and solve . We continue on using this same method where we increase the perfect square until can be made bigger than it. When we repeat this until we have rows, we get:Hence the solution is the last digit of , which is .。
大学生英语竞赛(NECCS)B类英语专业决赛真题2010年
大学生英语竞赛(NECCS)B类英语专业决赛真题2010年Part Ⅰ Listening C o m p e r h e n s i o n略Part Ⅱ Multiple C h o iceSection AThere are 10 incomplete sentences in this .section. For each blank there are four choices marked A, B, C and D. Choose the one that best completes the sentence. Then mark the corresponding letter on the answer sheet with a single line through the centre.1. A (n) gives you definite and arrival dates, and a list of all your destinations.A.timetable ; partingB.scheme ; leaveC.agenda ; quittingD.itinerary ; departureD 句意：一份行程表会告知你确定的出发和到达日期和你所有的目的地的列表。
itinerary行程，路线。
itinerary指行程表，旅行的时间安排。
timetable本身没有安排或计划的意思，可以是一些固定的时间表，常常针对的是一系列事件的时间的细节，如列车时刻表railway timetable。
scheme指计划，方案，不涉及时间概念。
agenda是会议的议程，待办事项表。
departure离开，出发，常见于火车等交通工具。
parting分手，分离，常指情侣间的分手或死亡造成的分离。
quitting放弃。
由题意可知，本题与旅行相关，所以选D。
2. year my travel agent promises me that my holiday will be best I have ever had, but none of these promises has ever come true.A.Every; theB.Each; ast; theD.Once a; aA 句意：每年我的旅行中介都向我保证我的假期将会是最好的一个，但是这些承诺从没实现过。
2010年全国中学生英语能力竞赛决赛答案
2010年全国中学生英语能力竞赛（NEPCS）决赛高二年级组试题参考答案听力部分（共三大题，计30分）I．Responses(句子应答)1－5ACBDAII．Dialogues(对话理解)A）6－10DCBAC B）11－15BCDCA C）16－20BCDAE III．Passages(短文理解)A)21-25ACBCDB)26. last week 27. close to 28. dangerous 29. 50,000 30. underground cave笔试部分（共七大题，计120分）I．Multiple choice(选择填空)31－35BCCDA36－40CADCB41－45DCABCII．Cloze(完形填空)46. take 47. went 48. himself 49. impossible 50. something 51. saw52. wanting 53. Within 54. managed 55. instantlyIII．Reading comprehension(阅读理解)56. address list 57. account 58. more careful 59. trusts 60. cheated 61. F 62. F63. 3:30 pm.64. No, he/she must be with a grown-up / someone over 18 years old.65. They should contact the General Office or visit the park’s website.66. B67. 所有的这些都是自动发生的，不容你考虑。
68. 那种情况发生在眼睛的前后径距离非常近的时候。
69. nearsighted / myopic70. inward; natural lens71－75FDBECIV．Translation(翻译)76. This rule does not apply in this case.77. Does the idea of studying abroad appeal to you?78. Once she gets into teaching, she’ll enjoy / like it.79. It was last autumn that I entered this senior high school.80. She is a famous professor, which is known to all of us / which we all know. V．IQ(智力测试)81. 142.The differences are: 1X19 2X19 3X19 4X19 5X19 6X1919 38 57 76 95 11482. peace83. Simone. (Names begin and end with the second and the second-to-last of the preceding name.)84.D85. $135. (Al’s share = $60 = 4 parts; Each part = $15; Original amount $15X9 = 135)VI．Error correction(短文改错)1. Thousands of trees were burning----burned2. one of the worst forest fire---- fires3. in Saturday afternoon---- on4. due of high winds---- to5. tried to bring---- trying6. total---- totally7. so a large and dark cloud---- such8. and fortunately---- but9. Rescue workers picked casualties---- Rescue workers picked up casualties (即加up)10. have been lost their homes---- have lost their homes (即去掉been)VII．Writing(写作)A) One possible version:A car was driving along a road, and a person was riding his bike along the same road. Suddenly, a cat ran into the road. The cyclist turned left to avoid hitting it and was hit by the passing car. The cyclist fell off his bike, seriously hurt and bleeding. The driver of the car jumped out of his car and dialed the emergency number. Soon an ambulance arrived and the attendants lifted the injured cyclist into the ambulance. B) One possible version:As a senior two student, I’m ver y busy with my studies. Sometimes I get really tired of studying, and I surf the Internet as a way to relax. However, my parents are strongly against my doing this. In their opinion, I should focus totally on my studies. Besides, they think that there is a lot of negative material on the Internet, and worry that I may become addicted to playing games. I can appreciate their concern, but I am not a kid, and know what I can and can’t do. In fact, I usually look at current affairs sites, sports activities and other news or pictures which interest me. I think it’s time totalk this over again with my parents. By telling them what I’ve learned from the Internet, I think we can reach an agreement.一、写作评分原则：1．本题总分为30分：A）10分；B）20分。
2010AMC10美国数学竞赛B卷
2010 AMC10美国数学竞赛B卷1. What is 10010031001003（）（）?---(A) -20,000 (B) -10,000 (C) -297 (D) -6 (E) 02. Makarla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?(A) 15 (B) 20 (C) 25 (D) 30 (E) 353. A drawer contains red, green, blue, and white socks with at least 2 of each color, What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?(A) 3 (B) 4 (C) 5 (D) 8 (E) 94. For a real number x, define ♡(x) to be the average of x and x2. What is♡(1)+ ♡(2)+ ♡(3)?(A) 3 (B) 6 (C) 10 (D) 12 (E) 205. A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?(A) 2 (B) 3 (C) 4 (D) 5 (E) 66. A circle is centered at O, AB is a diameter and C is a point on the circle with∠COB=50°. What is the degree measure of ∠CAB?(A) 20 (B) 25 (C) 45 (D) 50 (E) 657. A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?(A) 16 (B) 24 (C) 28 (D) 32 (E) 368. A ticket to a school play cost x dollars, where x is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for x are possible?(A) 1 (B) 2 (C) 3 (D) 4 (E) 59. Lucky Larry’s teacher asked him to substitute numbers for a, b, c, d, and e in the expression ((()))---+and evaluate the result. Larry ignored the parent thesea b c d ebut added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for a, b, c, and d were 1, 2, 3, and 4, respectively. What number did Larry substitute for e?(A) -5 (B) -3 (C) 0 (D) 3 (E) 510. Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rainin the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?(A) 18 (B) 21 (C) 24 (D) 27 (E) 3011. A shopper plans to purchase an item that has a listed price greater than $100 and can use any one of the three coupons. Coupon A gives 15% off the listed price, Coupon B gives $30 off the listed price, and Coupon C gives 25% off the amount by which the listed price exceeds $100.Let x and y be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is y-x?(A) 50 (B) 60 (C) 75 (D) 80 (E) 10012. At the beginning of the school year, 50% of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and 50% answered “No”. At the end of the school year, 70% answered “Yes” and 30% answered “No”. Altogether, x% of the students gave a difference between the maximum and the minimum possible values of x?(A) 0 (B) 20 (C) 40 (D) 60 (E) 8013. What is the sum of all the solutions of 2602=--?x x x(A) 32 (B)60 (C)92 (D) 120 (E) 12414. The average of the numbers 1, 2, 3, …, 98, 99, and x is 100x. What is x? (A)49101 (B) 50101 (C) 12 (D) 51101 (E) 509915. On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse ’s total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?(A) 25(B) 27 (C) 29 (D) 31 (E) 3316. A square of side length 1 and a circle of radiusshare the same center. What is the area inside the circle, but outside the square?(A)13π- (B) 29π- (C) 18π (D) 14(E) 29π17. Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea ’s score was the median among all students, and hers was the highest score on here team. Andrea ’s teammates Beth and Carla placed 37th and 64th , respectively. How many schools are in the city?(A) 22(B) 23 (C) 24 (D) 25 (E) 2618. Positive integers a, b, and c are randomly and independently selected withreplacement from the set {1, 2, 3, …, 2010}. What is the probability that abc ab a++ is divisible by 3?(A) 13(B) 2981(C) 3181(D) 1127(E) 132719. A circle with center O has area 156π. Triangle ABC is equilateral, BC is a chordon the circle, OA=, and point O is outside △ABC. What is the side length of △ABC?(A)(B) 64 (C) (D) 12 (E) 1820. Two circles lie outside regular hexagon ABCDEF. The first is tangent to AB, and the second is tangent to DE, Both are tangent to lines BC and FA. What is the ratio of the area of the second circle to that of the first circle?(A) 18 (B) 27 (C) 36 (D) 81 (E) 10821. A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?(A) 1/10 (B) 1/9 (C) 1/7 (D) 1/6 (E) 1/522. Seven distinct pieces of candy are to be distributed among three bags. The red bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?(A) 1930 (B) 1931 (C) 1932 (D) 1933 (E) 193423. The entries in a 3×3 array include all the digits from 1 through 9, arranged so thatthe entries in every row and column are in increasing order. How many such arrays are there?(A) 18(B) 24 (C) 36 (D) 42 (E) 6024. A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?(A) 30(B) 31 (C) 32 (D) 33 (E) 3425. Let a>0, and let P(x) be a polynomial with integer coefficients such that(1)(3)(5)(7),(2)(4)(6)(8).P P P P a and P P P P a ========- What is the smallest possible value of a?(A) 105(B) 315 (C) 945 (D) 7! (E) 8!。
2010年大学生英语竞赛(NECCS)B类决赛真题试卷(精选)(题后含答案及解析)
2010年大学生英语竞赛（NECCS）B类决赛真题试卷（精选）(题后含答案及解析)题型有：1. Within two years of the launch of Tiangong-1, China will launch shenzhou-VDI, Shenzhou-IX and Shenzhou-X spaceships, to dock with Tiangong-1, the website said. Two space laboratories, Tiangong- E and Tiangong- HI, will follow, and china aims to build its own space station by the year 2020, the website said. China became the third nation—after the US and Russia—to send people into space when Yang Li Wei went into orbit aboard the spaceship shenzhou-V on October 15, 2003. Three other astronauts were sent into space in Shenzhou-VH and carried out the country’s first space walk in September 2008. Shen Liping, deputy chief designer of China’s manned space programme, was quoted by the Guangzhou Daily as saying that China’s first woman astronaut will go into outer space sooner than the targeted 10 years from now.31．Chang’e-2, China’s second lunar probe, will be launched in October 2010, according to the chief designer of the nation’s first lunar probe.正确答案：Y解析：根据第三段最后一句Ye Peijian，…had told China daily earlier that the launch was expected in October．可知，中国首个月球探测器的总设计师叶培建向《中国13报》透露，预期在十月份发射嫦娥二号。
2010年大学生英语竞赛(NECCS)B类初赛真题试卷(题后含答案及解析)
2010年大学生英语竞赛（NECCS）B类初赛真题试卷(题后含答案及解析)题型有：1. Listening Comperhension 2. Multiple Choice 3. Cloze 4. Reading Comperhension 5. Translation 6. IQ Test 7. WritingPart I Listening ComperhensionSection A听力原文：M: I’ve just had a call from the bakers. They won’ t be able to finish the cake this afternoon.We’ll have to pick it up tomorrow morning. Who is going to pick it up? W: Shall I get it? Andy could take me on his motorbike. M: My wedding cake on the back of Andy’ s motorbike? No way!1．For what occasion has the cake been made?A．A weddingB．A birthday partyC．A conference reception正确答案：A解析：女士说My wedding cake on the back…可知蛋糕是结婚用的。
听力原文：W: Good morning. Can I help you?M: Yes, have you got any brochures on Africa? I am a keen photographer and I’d like to spend some time photographing wild animals. W: Well, I think we have something here to suit you. Let’ s see. We have two weeks in Kenya. It looks very attractive. I don’ t think you will be disappointed. They also guarantee plenty of wild life.2．Where does this conversation probably take place?A．A travel agencyB．A photography classC．A wildlife park正确答案：A解析：根据女士的话We have two weeks in Kenya…They also guarantee plenty of wild life可推测这是在一个旅行社，女士在向男士建议旅行项目。
AMC 10B 试题及答案解析
2010 AMC 10B1 . What is ?SolutionWe first expand the first term, simplify, and then compute to get an answer of .2 、Makarla attended two meetings during her -hour work day. The first meeting took minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?SolutionThe percentage of her time spent in meetings is the total amount of time spent in meetings divided by the length of her workday.The total time spent in meetings isTherefore, the percentage is3、A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?SolutionAfter you draw socks, you can have one of each color, so (according to the pigeonhole principle), if you pull then you will be guaranteed a matching pair.4 、For a real number , define to be the average of and . What is ?SolutionThe average of two numbers, and , is defined as . Thus the average of and would be . With that said, we need to find the sum when we plug, , and into that equation. So:.5 、A month with days has the same number of Mondays and Wednesdays.How many of the seven days of the week could be the first day of this month?Solution（B）. 36 、A circle is centered at , is a diameter and is a point on the circle with . What is the degree measure of ?SolutionAssuming the reader is not readily capable to understand how will always be right, the I will continue with an easily understandable solution. Since is the center, are all radii, they are congruent. Thus, and are isosceles triangles. Also, note thatand are supplementary, then . Since is isosceles, then . They also sum to , so each angle is .7 、A triangle has side lengths , , and . A rectangle has width and area equal to the area of the rectangle. What is the perimeter of this rectangle?SolutionThe triangle is isosceles. The height of the triangle is therefore given byNow, the area of the triangle isWe have that the area of the rectangle is the same as the area of the triangle, namely 48. We also have the width of the rectangle: 4.The length of the rectangle therefor is:The perimeter of the rectangle then becomes:The answer is:8 、A ticket to a school play cost dollars, where is a whole number. A group of 9th graders buys tickets costing a total of $, and a group of 10th graders buys tickets costing a total of $. How many values for are possible?SolutionWe see how many common integer factors 48 and 64 share. Of the factors of 48 - 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48; only 1, 2, 4, 8, and 16 are factors of 64. So there are possibilities for the ticket price.9 、Lucky Larry's teacher asked him to substitute numbers for , , , , and in the expression and evaluate the result. Larry ignored the parenthese but added and subtracted correctly andobtained the correct result by coincidence. The number Larry sustitued for , , , and were , , , and , respectively. What number did Larry substitude for ?SolutionSimplify the expression . I recommend to start with the innermost parenthesis and work your way out.So you get:Henry substituted with respectively.We have to find the value of , such that(the same expression without parenthesis).Substituting and simplifying we get:So Henry must have used the value for .Our answer is:10、Shelby drives her scooter at a speed of miles per hour if it is not raining, and miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of miles in minutes. How many minutes did she drive in the rain?SolutionWe know thatSince we know that she drove both when it was raining and when it was not and that her total distance traveled is miles.We also know that she drove a total of minutes which is of an hour. We get the following system of equations, where is the time traveled when it was not raining and is the time traveled when it was raining:Solving the above equations by multiplying the second equation by 30 and subtracting the second equation from the first we get:We know now that the time traveled in rain was of an hour, which is minutesSo, our answer is:11 、A shopper plans to purchase an item that has a listed price greater than $and can use any one of the three coupns. Coupon A givesoff the listed price, Coupon B gives $off the listed price, and Coupon C gives off the amount by which the listed price exceeds $.Let and be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is −?SolutionLet the listed price be , whereCoupon A saves us:Coupon B saves us:Coupon C saves us:Now, the condition is that A has to be greater than or equal to either B or C which give us the following inequalities:We see here that the greatest possible value for p is and the smallest isThe difference between and isOur answer is:12 、At the beginning of the school year, of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", andanswered "No." At the end of the school year, answered "Yes" and answerws "No." Altogether, of the students gave a differentanswer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of ?SolutionThe minimum possible value occurs when of the students who originally answered "No." answer "Yes." In this case,The maximum possible value occurs when of the students who originally answered "Yes." answer "No." and the of the students who originally answered "No." answer "Yes." In this case,Subtract to obtain an answer of13 、What is the sum of all the solutions of ?SolutionCase 1:Case 1a:Case 1b:Case 2:Case 2a:Case 2b:Since an absolute value cannot be negative, we exclude . The answer is14 、The average of the numbers and is . What is ?SolutionWe must find the average of the numbers from to and in terms of . The sum of all these terms is . We must divide this by the total number of terms, which is . We get: . This is equal to , as stated in the problem. We have: . We can now cross multiply. This gives:This gives us our answer.15 、On a -question multiple choice math contest, students receive points for a correct answer, points for an answer left blank, and point for an incorrect answer. Jesse’s total score on the contest was . What is the maximum number of questions that Jesse could have answered correctly?SolutionLet be the amount of questions Jesse answered correctly, be the amount of questions Jesse left blank, and be the amount of questions Jesse answered incorrectly. Since there were questions on the contest, . Since his total score was , . Also,. We can substitute this inequality into the previous equation to obtain another inequality:. Since is an integer, the maximum value for is .16 、A square of side length and a circle of radius share the same center. What is the area inside the circle, but outside the square?Solution(B)17 、Every high school in the city of Euclid sent a team of students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed th and th, respectively. How many schools are in the city?SolutionLet the be the number of schools, be the number of contestants, and be Andrea's score. Since the number of participants divided by three isthe number of schools, . Andrea received a higher score than her teammates, so . Since is the maximum possible median, then is the maximum possible number ofparticipants. Therefore, . This yields thecompound inequality: . Since a set with an even number of elements has a median that is the average of the two middle terms, an occurrence that cannot happen in this situation, cannot be even.is the only other option.18 、Positive integers , , and are randomly and independently selected with replacement from the set . What is the probability that is divisible by ?Solution（E）13/2719 、A circle with center has area . Triangle is equilateral,is a chord on the circle, , and point is outside . What is the side length of ?Solution(B)620 、Two circles lie outside regular hexagon . The first is tangent to , and the second is tangent to . Both are tangent to lines and . What is the ratio of the area of the second circle to that of the first circle?SolutionA good diagram here is very helpful.The first circle is in red, the second in blue. With this diagram, we can see that the first circle is inscribed in equilateral triangle while the second circle is inscribed in . From this, it's evident that the ratio of the red area to the blue area is equal to the ratio of the areas of triangles toSince the ratio of areas is equal to the square of the ratio of lengths, weknow our final answer is From the diagram, we can see that this isThe letter answer is D21 、A palindrome between and is chosen at random. What is the probability that it is divisible by ?SolutionView the palindrome as some number with form (decimal representation): . But because the number is a palindrome, . Recombining this yields . 1001 is divisible by 7, which means that as long as , the palindrome will be divisible by 7. This yields 9 palindromes out of 90 () possibilities for palindromes. However, if , then this givesanother case in which the palindrome is divisible by 7. This adds another 9 palindromes to the list, bringing our total to22 、Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?SolutionWe can count the total number of ways to distribute the candies (ignoring the restrictions), and then subtract the overcount to get the answer.Each candy has three choices; it can go in any of the three bags.Since there are seven candies, that makes the total distributionsTo find the overcount, we calculate the number of invalid distributions: the red or blue bag is empty.The number of distributions such that the red bag is empty is equal to , since it's equivalent to distributing the 7 candies into 2 bags.We know that the number of distributions with the blue bag is empty will be the same number because of the symmetry, so it's also .The case where both the red and the blue bags are empty (all 7 candies are in the white bag) are included in both of the above calculations, and this case has only distribution.The total overcount isThe final answer will beThat makes the letter choice C23 、The entries in a array include all the digits from through ,arranged so that the entries in every row and column are in increasing order. How many such arrays are there?Solution（D）6024 、A high school basketball game between the Raiders and Wildcatswas tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than points. What was the total number of points scored by the two teams in the first half?SolutionRepresent the teams' scores as: andWe have Manipulating this, we can get, orSince both are increasing sequences, . We can check cases up to because when , we get . When▪▪▪Checking each of these cases individually back into the equation, we see that only when a=5 and n=2, we get an integer value for m, which is 9. The original question asks for the first half scores summed, so we must find25 、Let , and let be a polynomial with integer coefficients such that, and.What is the smallest possible value of ?SolutionThere must be some polynomial such thatThen, plugging in values of we getThus, the least value of must be the . Solving, we receive , so our answer is .。
2010 AMC10A试题(含答案)
2010 AMC10A ProblemsQ1. Mary’s top book shelf holds five books with the follo wing widths, in centimeters:6,12, 1, 2.5, and 10. What is the average book width, in centimeters?A) 1 B) 2 C) 3 D) 4 E)5Q2. Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?A)54B)43C)32D) 2 E)3Q3.Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?A)3 B)13 C)18 D) 25 E)29 Q4.A book that is to be recorded onto compact discs takes 412 minutes to read aloud. Each disc can hold up to 56 minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?A)50.2 B)51.5 C)52.4 D) 53.8 E)55.2Q5.The area of a circle whose circumference is 24π is kπ. What is the value of k?A)6 B)12 C)24 D) 36 E)144Q6. For positive numbers x and y the operation ♠(x, y) is defined as1(,)x y xy♠=-What is ♠(2, ♠(2, 2))?A)23B)1 C)43D)53E)2Q7.Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run?A)1 B) D)2 E)Q8.Tony works 2 hours a day and is paid $0.50 per hour for each full year of his age. During a six month period Tony worked 50 days and earned $630. How old was Tony at the end of the six month period?A)9 B)11 C)12 D) 13 E)14Q9.A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x + 32 are three -digit and four -digit palindromes, respectively. What is the sum of the digits of x ?A)20 B)21 C)22 D) 23 E)24Q11.The length of the interval of solutions of the inequality 23a x b ≤+≤is 10.What is b a -？A)6 B)10 C)15 D) 20 E)30Q12.Logan is constructing a scaled model of his town. The city’s water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan’s miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?A)0.04 B)0.4π C)0.4 D) 4πE)4Q13.Angelina drove at an average rate of 80 kmh and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kmh. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time t in hours that she drove before her stop? A)880100()2503t t +-= B)80250t = C)100250t = D)90250t = E)880()1002503t t -+=Q14.Triangle ABC has 2AB AC =⋅. Let D and E be on AB and BC , respectively, such that ∠BAE = ∠ACD . Let F be the intersection of segments AE and CD , and suppose that △CFE is equilateral. What is ∠ACB ?A)60° B)75° C)90° D) 105° E)120°Q15.In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.Brian: “Mike and I are different species.”Chris: “LeRoy is a frog.”LeRoy: “Chris is a frog.”Mike: “Of the four of us, at least two are toads.”How many of these amphibians are frogs?A)0 B)1 C)2 D) 3 E)4Q16.Nondegenerate △ABC has integer side lengths, BD is an angle bisector, AD = 3, and DC = 8. What is the smallest possible value of the perimeter?A)30 B)33 C)35 D) 36 E)37 Q17.A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?A)7 B)8 C)10 D) 12 E)15Q18.Bernardo randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set {1, 2, 3, 4, 5, 6, 7, 8} and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo’s number is larger than Silvia’s number?A)4772B)3756C)23D)4972E)3956Q19.Equiangular hexagon ABCDEF has side lengths AB = CD = EF = 1 and BC = DE = F A = r. The area of △ACE is 70% of the area of the hexagon. What is the sum of all possible values of r?A)3B)103C)4 D)174E) 6Q20.A fly trapped inside a cubical box with side length 1 meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?A)4+B)2+C)2+D)E)Q21.The polynomial 322010-+-has three positive integer roots. What isx ax bxthe smallest possible value of a?A)78 B)88 C)98 D)108 E)118Q22.Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?A) 28 B) 56 C) 70 D) 84 E) 140Q23.Each of 2010 boxes in a line contains a single red marble, and for 1 ≤k≤2010, the box in the k th position also contains k white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let P(n) be the probability that Isabella stops after drawing exactly n marbles. What is the smallest value of n for which1()P n<?2010A) 45 B) 63 C) 64 D) 201 E) 1005Q24.The number obtained from the last two nonzero digits of 90! is equal to n. What is n?A) 12 B) 32 C) 48 D) 52 E) 68Q25.Jim starts with a positive integer n and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with n = 55, then his sequence contains 5 numbers:Let N be the smallest number for which Jim’s sequence has 8 numbers. What is the units digit of N?A) 1 B) 3 C) 5 D) 7 E) 9AMC10A 2010中文解析Q1. Mary’s top book shelf holds five books with the follo wing widths, in centimeters:6,12, 1, 2.5, and 10. What is the average book width, in centimeters?A) 1 B) 2 C) 3 D) 4 E)5翻译：Mary最上面的书架上有五本书，宽度分别为: 6厘米、1厘米、2.5厘米和10厘米。
-年amc 10a和b竞赛真题及答案(英文版)
2010-2015年AMC 10A和B竞赛真题及答案（英文版）
你好，目前，只分享了2010-2013年AMC 10A和B竞赛真题及答案（英文版），2014-2015年的暂时还没有上传，等采纳后再私信我吧！需要的话，我可以将Word文档中的原图文件一同发给你，求采纳，毕竟我花了差不多一个下午才整理完，谢谢！(@_@)
2010年AMC 10Aห้องสมุดไป่ตู้赛真题及答案（英文版）
2010年AMC 10B竞赛真题及答案（英文版）
2011年AMC 10A竞赛真题及答案（英文版）
2011年AMC 10B竞赛真题及答案（英文版）
Problem 4
2012年AMC 10A竞赛真题及答案（英文版）
go去wentgone
2012年AMC 10B竞赛真题及答案（英文版）
get得到gotgot
sink下沉sank / sunksunk / sunken
give给gavegiven
hit打hithit
不规则动词表
2013年AMC 10A竞赛真题及答案（英文版）
mistake误认mistookmistaken
grow成长grewgrown
think思考thoughtthought
AMC10的真题答案及中文翻译
AMC10的真题及中文翻译1、One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?(A) 2 (B) 5 (C) 10 (D) 15 (E) 20中文：一张展览票全价为20美元。
Susan用优惠券买4张票打七五折。
Pam用优惠券买5张票打七折。
Pam比Susan多花了多少美元？2、An aquarium has a rectangular base that measures 100cm by 40cm and has a height of 50cm. It is filled with water to a height of 40cm. A brick with a rectangular base that measures 40cm by 20cm and a height of 10cm is placed in the aquarium. By how many centimeters does that water rise?(A) 0.5 (B) 1 (C) 1.5 (D) 2 (E)2.5中文：一个养鱼缸有100cm×40cm的底，高为50cm。
它装满水到40cm的高度。
把一个底为40cm×20cm，高为10cm的砖块放在这个养鱼缸里。
鱼缸里的水上升了多少厘米？3、The larger of two consecutive odd integers is three times the smaller. What is their sum?(A) 4 (B) 8 (C) 12 (D) 16 (E) 20中文：2个连续的奇整数中较大的数是较小的数的3倍。
amc10数学竞赛题目
amc10数学竞赛题目
以下是一道AMC 10数学竞赛题目的例子：
问题：在平面上，有一个边长为10的正方形ABCD，点E是AB
的中点，点F是BC的中点。
连接线段AF和BE，交于点P。
求AP的长度。
A) 2
B) 4
C) 5
D) 6
E) 8
解析：
首先，我们可以观察到三角形APE与三角形FPC共有一个角A，因此它们是相似三角形。
根据相似三角形的性质，我们可以得出以下比例关系：AP/FP = AE/FC = 1/2。
另一方面，我们可以利用勾股定理来计算AE和FC的长度。
由于AE是AB的中点，而AB的长为10，所以AE的长度为10/2 = 5。
同样地，FC是BC的中点，而BC的长也是10，所以FC的长度也是10/2 = 5。
将上面的结果代入比例关系中，我们得到AP/FP = 5/5 = 1。
因此，AP和FP的长度是相等的。
最后，我们可以发现三角形APF是一个等边三角形，因此AP的长度等于FP的长度，即AP = FP = 5。
所以，答案是C) 5。