AMC10美國數(shù)學競賽真題2002A卷.doc

1、Problem 11020OC + 102002The ratio1020。1+ 1020 is closest to which of the following numbers?(A) 0.1(B) 0.2(C) 1(D) 5(E) 10SolutionProblem 2Given that a, b, and c are non-zero real numbers, defineFind (2, 12, 9).(A) 4(B) 5(C) 6(D) 7(E) 8SolutionProblem 3According to the standard convention for exponen

2、tiation,=216 = 65,536.SolutionIf the order in which the exponentiations are performed is changed, how many other values are possible?(A) 0(B) 1(C) 2(D) 3(E) 4SolutionProblem 4For how many positive integers mn < m + n?mis there at least 1 positive integer M such that(A) 4(B) 6(C) 9(D) 1 2(V.!infin

3、itely manySolutionProblem 5Each of the small circles in the figure has radius one. The innermost cwcle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.(A) 7T (B) 1.57T(C) 27

4、r(D) 37r(E) 357rSolutionProblem 6From a starting number, Cindy was supposed to subtract 3, and then divide by 9, but instead, Cindy subtracted 9, then divided by 3, getting 43. If the correct instructions were followed, what would the result be?(A) 15(B) 34(C) 43(D) 51(E)138SolutionProblem 7A 15 arc

5、 of cii*cle A is equal in length to a 300arc of circle B. What is the ratio of circle A's area and circle B's area?(A) 4/9(B) 2/3(C) 5/6(D) 3/2(E) 9/4SolutionProblem 8Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let Ube the total area of

6、 the blue triangles, IT the total area of the white squares, and /?the area of the red square. Which of the following is correct?(A) B = W (B) W = R (C) B=B (D) 3B = 2R (E) 2R = WProblem 9There are 3 numbers A, B, and C, such that 1001C - 2002.4 = 4004, and 10018 + 3003A = 5005.What is the average o

7、f A, B, and C?(-A) 1(R) 3(C) 6(D) 9(r.)Not uniquely determinedSolutionProblem 10What is the sum of all of the roots of(2t + 3)(z - 4) + (2x + 3)(1 - 6) = 0?(A) 7/2(B) 4(C) 5(D) 7(E) 13SolutionProblem 11Jamal wants to save 30 files onto dis ks, each with 1.44 MB space. 3 of the files take up 0.8 MB e

8、ach, 12 of the files take up 0.7 MB each, and the rest take up 0.4 MB each. It is not possible to split a file onto 2 different disks. What is the smallest number of dis ks needed to store all 30 files?(A) 12(B) 13(C) 14(D) 15(E)16SolutionProblem 12Mr. Earl E. Bird leaves home every day at 8:00 AM t

9、o go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?(A) 45(B) 48(C) 50(D) 55(E)58Sol

10、utionProblem 13Given a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.(A) 6(B) 12(C) 12.5(D) 13(E) 15SolutionProblem 14Both roots of the quadratic equation 1廣 63/ + A： = Oare prime numbers. The number of possible values of Ris(A) 0(B) 1(C) 2(D)4(E) more than 4Solu

11、tionProblem 15Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?(A) 150(B) 160(C) 170(D) 180(E) 19()SolutionProblem 16Let a + 1 =匕 + 2 = c + 3 = d + 4 = a + 匕 + c + d + 5. What is (1 + b + c + d ?(A) - 5(B)

12、 - 10/3(C) - 7/3(D) 5/3(E) 5SolutionProblem 17Sarah pours 4 ounces of coffee into a cup that can hold 8 ounces. Then she pours 4 ounces of cream into a second cup that can also hold 8 ounces. She then pours half of the contents of the first cup into the second cup, completely mixes the contents of t

13、he second cup, then pours half of the contents of the second cup back into the first cup. What fraction of the contents in the cup is cream?(A) 1/4(B) 1/3(C) 3/8(D) 2/5(E)l/2SolutionProblem 18A 3x3x3 cube is made of 27 normal dice. Each die's opposite sides sum to 7.What is the smallest possible

14、 sum of all of the values vis ible on the 6 foces of the large cube?(A) 60(D) 72(C) 84(D) 90(E)96SolutionProblem 19SolutionSpot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the

15、region outs ide of the doghouse that Spot can reach?(A) 2?r/3(B) 27r(C) 5”2(D) 8tt/3(E) 3喬SolutionProblem 20Points,，"and 廠 lie, in that order, ondividing it into fivesegments, each oflength 1. Point (7is not on line AF . Point II lies on Cr D ,and point J lies on GF. The line segmentsJ_Eand &qu

16、ot;Gare parallel.Find 1IC/JE(A) 5/4(B) 4/3(C) 3/2(D) 5/3(E) 2SolutionProblem 21The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is (A) 11(B) 12(C) 13(D) 11(E) 15SolutionProblem 22A set of tile

17、s numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square , and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?(A) 10(B) 11(C)

18、 18(D) 19(E) 20SolutionProblem 23Points 兒區(qū) Gind Dlie on a line, in that order, with= COand.Pointline, ani? . The perimeter of BE = CE = 10Dis twice the perimeter of. Find AB.(A) 15/2(B)8(C) 17/2(D) 9(E) 19/2Problem 24Tina randomly selects two distinct numbers from the set 1, 2, 3, 4, 5, and Sergio randomly selects a number from the set 1, 2,，10. What is the probability