ΒΆ 2002 AMC 12A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2002 AMC 12A problems here.
Discussion Forum
Engage in discussion about the 2002 AMC 12A math contest by visiting Random Math AMC 12A 2002 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2002 AMC 12A problems, please refer below:
Problem 1: Compute the sum of all the roots of .
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Problem 2: Cindy was asked by her teacher to subtract from a certain number and then divide the result by . Instead, she subtracted and then divided the result by , giving an answer of . What would her answer have been had she worked the problem correctly?
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Problem 3: According to the standard convention for exponentiation,
If the order in which the exponentiations are performed is changed, how many other values are possible?
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Problem 4: Find the degree measure of an angle whose complement is of its supplement.
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Problem 5: Each 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.
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Problem 6: For how many positive integers does there exist at least one positive integer such that ?
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Problem 7: If an of on circle has the same length as an of on circle , then the ratio of the area of circle to the area of circle is
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Problem 8: Betsy designed a flag using blue triangles (β ), small white squares , and a red center square (β ), 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?
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Problem 9: Jamal wants to store computer files on floppy disks, each of which has a capacity of megabytes (mb). Three of his files require of memory each, more require each, and the remaining require each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?
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Problem 10: Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
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Problem 11: Mr. Earl E. Bird leaves his house for work at exactly A.M. every morning. When he averages miles per hour, he arrives at his workplace three minutes late. When he averages miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?
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Problem 12: Both roots of the quadratic equation are prime numbers. The number of possible values of is
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Problem 13: Two different positive numbers and each differ from their reciprocals by . What is ?
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Problem 14: For all positive integers , let . Let
Which of the following relations is true?
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Problem 15: The mean, median, unique mode, and range of a collection of eight integers are all equal to . The largest integer that can be an element of this collection is
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Problem 16: Tina randomly selects two distinct numbers from the set , and Sergio randomly selects a number from the set . The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is
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Problem 17: Several 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?
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Problem 18: Let and be circles defined by
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Problem 19: The graph of the function is shown below. How many solutions does the equation have?
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Problem 20: Suppose that and are digits, not both nine and not both zero, and the repeating decimal is expressed as a fraction in lowest terms. How many different denominators are possible?
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Problem 21: Consider the sequence of numbers: For , the th term of the sequence is the units digit of the sum of the two previous terms. Let denote the sum of the first terms of this sequence. The smallest value of for which is:
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Problem 22: Triangle is a right triangle with as its right angle, , and . Let be randomly chosen inside , and extend to meet at . What is the probability that ?
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Problem 23: In triangle , side and the perpendicular bisector of meet in point , and bisects . If and , what is the area of triangle ?
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Problem 24: Find the number of ordered pairs of real numbers such that .
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Problem 25: The nonzero coefficients of a polynomial with real coefficients are all replaced by their mean to form a polynomial . Which of the following could be a graph of and over the interval ?
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The problems on this page are the property of the MAA's American Mathematics Competitions