AMC10美国数学竞赛真题2002A卷.pdf

Problem 1The ratio is closest to which of the following numbers?SolutionProblem 2Given that a,b,and c are non-zero real numbers,define.Find.SolutionProblem 3According to the standard convention for exponentiation,.If the order in which the exponentiations are performed is changed,how many other values are possible?SolutionProblem 4For how many positive integers is there at least 1 positive integer such that?infinitely manySolutionProblem 5Each of the small circles in the figure has radius one.The innermost circle 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.SolutionProblem 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?SolutionProblem 7A arc of circle A is equal in length to a arc of circle B.What is the ratio of circle As area and circle Bs area?SolutionProblem 8Betsy designed a flag using blue triangles,small white squares,and a red center square,as shown.Let be the total area of the blue triangles,the total area of the white squares,and the area of the red square.Which of the following is correct?SolutionProblem 9There are 3 numbers A,B,and C,such that,and.What is the average of A,B,and C?Not uniquely determinedSolutionProblem 10What is the sum of all of the roots of?SolutionProblem 11Jamal wants to save 30 files onto disks,each with 1.44 MB space.3 of the files take up 0.8 MB each,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 disks needed to store all 30 files?SolutionProblem 12Mr.Earl E.Bird leaves home every day at 8:00 AM to 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?SolutionProblem 13Given a triangle with side lengths 15,20,and 25,find the triangles smallest height.SolutionProblem 14Both roots of the quadratic equation are prime numbers.The number of possible values of is SolutionProblem 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?SolutionProblem 16Let.What is?SolutionProblem 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 the second cup,then pours half of the contents of the second cup back into the first cup.What fraction of the contents in the first cup is cream?SolutionProblem 18A 3x3x3 cube is made of 27 normal dice.Each dies opposite sides sum to 7.What is the smallest possible sum of all of the values visible on the 6 faces of the large cube?SolutionProblem 19Spots 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 region outside of the doghouse that Spot can reach?SolutionProblem 20Points and lie,in that order,on,dividing it into five segments,each of length 1.Point is not on line.Point lies on,and point lies on.The line segments and are parallel.Find.SolutionProblem 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 isSolutionProblem 22A set of tiles 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?SolutionProblem 23Points and lie on a line,in that order,with and.Point is not on the line,and.The perimeter of is twice the perimeter of.Find.SolutionProblem 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 that Sergios number is larger than the sum of the two numbers chosen by Tina?SolutionProblem 25In trapezoid with bases and,we have,and(diagram not to scale).The area of isSolution