Problem Set Workbook
Access the downloadable workbook for 2005 AIME I problems is available here.
Discussion Forum
Engage in discussion about the 2005 AIME I math contest by visiting Random Math AIME I 2005 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2005 AIME I problems, please refer below:
Problem 1: Six congruent circles form a ring with each circle externally tangent to the two circles adjacent to it. All six circles are internally tangent to a circle with radius . Let be the area of the region inside and outside all of the six circles in the ring. Find . (The notation denotes the greatest integer that is less than or equal to .)
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Problem 2: For each positive integer , let denote the increasing arithmetic sequence of integers whose first term is and whose common difference is . For example, is the sequence For how many values of does contain the term
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Problem 3: How many positive integers have exactly three proper divisors, each of which is less than (A proper divisor of a positive integer is a positive integer divisor of other than itself.)
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Problem 4: The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are members left over. The director finds that if they are arranged in a rectangular formation with more rows than columns, the desired result can be obtained. Find the maximum number of members this band can have.
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Problem 5: Robert has indistinguishable gold coins and indistinguishable silver coins. Each coin has an engraving of a face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the coins.
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Problem 6: Let be the product of the nonreal roots of . Find . (The notation denotes the greatest integer that is less than or equal to .)
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Problem 7: In quadrilateral , and . Given that , where and are positive integers, find .
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Problem 8: The equation
has three real roots. Given that their sum is , where and are relatively prime positive integers, find .
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Problem 9: Twenty-seven unit cubes are each painted orange on a set of four faces so that the two unpainted faces share an edge. The cubes are then randomly arranged to form a cube. Given that the probability that the entire surface of the larger cube is orange is , where , and are distinct primes and , and are positive integers, find .
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Problem 10: Triangle lies in the Cartesian plane and has area . The coordinates of and are and , respectively, and the coordinates of are . The line containing the median to side has slope . Find the largest possible value of .
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Problem 11: A semicircle with diameter is contained in a square whose sides have length . Given that the maximum value of is , where and are integers, find .
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Problem 12: For positive integers , let denote the number of positive integer divisors of , including and . For example, and . Define by
Let denote the number of positive integers with odd, and let denote the number of positive integers with even. Find .
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Problem 13: A particle moves in the Cartesian plane from one lattice point to another according to the following rules:
From any lattice point , the particle may move only to , or .
There are no right angle turns in the particle's path. That is, the sequence of points visited contains neither a subsequence of the form ) nor a subsequence of the form .
How many different paths can the particle take from to
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Problem 14: Consider the points , and . There is a unique square such that each of the four points is on a different side of . Let be the area of . Find the remainder when is divided by .
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Problem 15: In . The incircle of the triangle divides the median containing into three segments of equal length. Given that the area of is , where and are integers and is not divisible by the square of any prime, find .
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The problems on this page are the property of the MAA's American Mathematics Competitions