美国数学竞赛amc12

2002 AMC 12A ProblemsProblem 1Compute the sum of all the roots ofProblem 2Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?Problem 3According to the standard convention for exponentiation,If the order in which the exponentiations are performed is changed, how many other values are possible?Problem 4Find the degree measure of an angle whose complement is 25% of its supplement.Problem 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.Problem 6For how many positive integers does there exist at least one positive integer n suchthat ?infinitely manyProblem 7A arc of circle A is equal in length to a arc of circle B. What is the ratio of circle A's area and circle B's area?Problem 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?Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. 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?Problem 10Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?Problem 11Mr. Earl E. Bird gets up 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?Problem 12Both roots of the quadratic equation are prime numbers. The number ofpossible values of isProblem 13Two different positive numbers and each differ from their reciprocals by . Whatis ?For all positive integers , let .Let . Which of the following relations is true?Problem 15The 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 isProblem 16Tina 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 Sergio's number is larger than the sum of the two numbers chosen by Tina?Problem 17Several sets of prime numbers, such as use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?Problem 18Let and be circles definedby and respectively. What is the length ofthe shortest line segment that is tangent to at and to at ?The graph of the function is shown below. How many solutions does theequation have?Problem 20Suppose that and are digits, not both nine and not both zero, and the repeatingdecimal is expressed as a fraction in lowest terms. How many different denominators are possible?Problem 21Consider the sequence of numbers: For , the -th term of thesequence is the units digit of the sum of the two previous terms. Let denote the sum of thefirst terms of this sequence. The smallest value of for which is:Problem 22Triangle is a right triangle with as its rightangle, , and . Let be randomly chosen inside ,and extend to meet at . What is the probability that ?Problem 23In triangle , side and the perpendicular bisector of meet in point ,and bisects . If and , what is the area oftriangle ?Problem 1In the year, the United States will host the International Mathematical Olympiad.Let and be distinct positive integers such that the product .What is the largest possible value of the sum ?Problem 2Problem 3Each day, Jenny ate of the jellybeans that were in her jar at the beginning of that day.At the end of the second day, remained. How many jellybeans were in the jar originally?Problem 4The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?Problem 5If where thenProblem 6Two different prime numbers between and are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?Problem 7How many positive integers have the property that is a positive integer?Problem 8Figures, , , and consist of , , , and non-overlapping squares. If thepattern continued, how many non-overlapping squares would there be in figure?Problem 9Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71,76,80,82, and 91. What was the last score Mrs. Walters entered?Problem 10The point is reflected in the -plane, then its image is rotatedby about the -axis to produce , and finally, is translated by 5 units in thepositive-direction to produce . What are the coordinates of ?Problem 11Two non-zero real numbers, and satisfy. Which of the following is apossible value of?Problem 12Let A, M, and C be nonnegative integers such that . What is the maximumvalue of + + + ?Problem 13One morning each member of Angela’s family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?Problem 14When the mean,median, and modeof the listare arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of?Problem 15Let be a function for which . Find the sum of all values of forwhich.Problem 16A checkerboard of rows and columns has a number written in each square, beginning inthe upper left corner, so that the first row is numbered , the secondrow , and so on down the board. If the board is renumbered so that the left column,top to bottom, is , the second column and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).Problem 17A centered at has radius and contains the point . The segment is tangent tothe circle at and . If point lieson and bisects , thenProblem 18In year , the day of the year is a Tuesday. In year , the day isalso a Tuesday. On what day of the week did th day of year occur?Problem 19triangle , , , . Let denote the midpointof and let denote the intersection of with the bisector of angle .Which of the following is closest to the area of the triangle ?Problem 20If and are positive numbers satisfyingThen what is the value of latex ?Problem 21Through a point on the hypotenuse of right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into asquare and two smaller right triangles. The area of one of the two small right triangles times the area of the square. The ratio of the area of the other small right triangle to the area of the square isProblem 22The graph below shows a portion of the curve defined by the quarticpolynomial. Which of the following is the smallest?Problem 23Professor Gamble buys a lottery ticket, which requires that he pick six different integersfrom through , inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property— the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket?Problem 24If circular arcs and centers at and , respectively, then there exists acircletangent to both and , and to . If the length of is , then the circumference of the circle isProblem 25Eight congruent Equilateral triangle each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)。