Problem Set Workbook
Access the downloadable workbook for 2003 AIME II problems here.
Discussion Forum
Engage in discussion about the 2003 AIME II math contest by visiting Random Math AIME II 2003 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2003 AIME II problems, please refer below:
Problem 1: The product of three positive integers is times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of .
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Problem 2: Let be the greatest integer multiple of , no two of whose digits are the same. What is the remainder when is divided by ?
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Problem 3: Define a good word as a sequence of letters that consists only of the letters , , and - some of these letters may not appear in the sequence - and in which is never immediately followed by is never immediately followed by , and is never immediately followed by . How many seven-letter good words are there?
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Problem 4: In a regular tetrahedron, the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is , where and are relatively prime positive integers. Find .
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Problem 5: A cylindrical log has diameter inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as , where is a positive integer. Find .
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Problem 6: In , and point is the intersection of the medians. Points , and are the images of , and , respectively, after a rotation about . What is the area of the union of the two regions enclosed by the triangles and
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Problem 7: Find the area of rhombus given that the radii of the circles circumscribed around triangles and are and , respectively.
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Problem 8: Find the eighth term of the sequence , whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
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Problem 9: Consider the polynomials and . Given that , and are the roots of , find .
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Problem 10: Two positive integers differ by . The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?
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Problem 11: Triangle is a right triangle with , and right angle at . Point is the midpoint of , and is on the same side of line as so that . Given that the area of can be expressed as , where , and are positive integers, and are relatively prime, and is not divisible by the square of any prime, find .
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Problem 12: The members of a distinguished committee were choosing a president, and each member gave one vote to one of the candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least than the number of votes for that candidate. What is the smallest possible number of members of the committee?
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Problem 13: A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is , where and are relatively prime positive integers, find .
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Problem 14: Let and be points on the coordinate plane. Let be a convex equilateral hexagon such that , , and the -coordinates of its vertices are distinct elements of the set . The area of the hexagon can be written in the form , where and are positive integers and is not divisible by the square of any prime. Find .
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Problem 15: Let
Let be the distinct zeros of , and let for , where , and and are real numbers. Let
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