Problem Set Workbook
Access the downloadable workbook for 2007 AMC 12A problems here.
Discussion Forum
Engage in discussion about the 2007 AMC 12A math contest by visiting Random Math AMC 12A 2007 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2007 AMC 12A problems, please refer below:
Problem 1: One ticket to a show costs at full price. Susan buys tickets using a coupon that gives her a discount. Pam buys tickets using a coupon that gives her a discount. How many more dollars does Pam pay than Susan?
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Problem 2: An aquarium has a rectangular base that measures by and has a height of . It is filled with water to a height of . A brick with a rectangular base that measures by and a height of is placed in the aquarium. By how many centimeters does the water rise?
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Problem 3: The larger of two consecutive odd integers is three times the smaller. What is their sum?
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Problem 4: Kate rode her bicycle for minutes at a speed of , then walked for minutes at a speed of . What was her overall average speed in miles per hour?
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Problem 5: Last year Mr. John Q. Public received an inheritance. He paid in federal taxes on the inheritance, and paid of what he had left in state taxes. He paid a total of for both taxes. How many dollars was the inheritance?
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Problem 6: Triangles and are isosceles with and . Point is inside , and . What is the degree measure of ?
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Problem 7: Let , and be five consecutive terms in an arithmetic sequence, and suppose that . Which of the following can be found?
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Problem 8: A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from to , from to , from to , and so on, ending back at . What is the degree measure of the angle at each vertex in the star-polygon?
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Problem 9: Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
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Problem 10: A triangle with side lengths in the ratio is inscribed in a circle of radius . What is the area of the triangle?
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Problem 11: A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with terms , and and end with the term . Let be the sum of all the terms in the sequence. What is the largest prime number that always divides ?
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Problem 12: Integers , and , not necessarily distinct, are chosen independently and at random from to , inclusive. What is the probability that is even?
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Problem 13: A piece of cheese is located at in a coordinate plane. A mouse is at and is running up the line . At the point the mouse starts getting farther from the cheese rather than closer to it. What is ?
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Problem 14: Let , and be distinct integers such that
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Problem 15: The set is augmented by a fifth element , not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of ?
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Problem 16: How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?
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Problem 17: Suppose that and . What is ?
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Problem 18: The polynomial has real coefficients, and . What is ?
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Problem 19: Triangles and have areas and , respectively, with , and . What is the sum of all possible -coordinates of ?
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Problem 20: Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?
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Problem 21: The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function are equal. Their common value must also be which of the following?
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Problem 22: For each positive integer , let denote the sum of the digits of . For how many values of is ?
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Problem 23: Square has area 36 , and is parallel to the -axis. Vertices , and are on the graphs of , and , respectively. What is ?
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Problem 24: For each integer , let be the number of solutions of the equation on the interval . What is
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Problem 25: Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of , including the empty set, are spacy?
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The problems on this page are the property of the MAA's American Mathematics Competitions