Problem Set Workbook
Access the downloadable workbook for 2010 AIME II problems here.
Discussion Forum
Engage in discussion about the 2010 AIME II math contest by visiting Random Math AIME II 2010 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2010 AIME II problems, please refer below:
Problem 1: Let be the greatest integer multiple of all of whose digits are even and no two of whose digits are the same. Find the remainder when is divided by .
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Problem 2: A point is chosen at random in the interior of a unit square . Let denote the distance from to the closest side of . The probability that is equal to , where and are relatively prime positive integers. Find .
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Problem 3: Let be the product of all factors (not necessarily distinct) where and are integers satisfying . Find the greatest positive integer such that divides .
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Problem 4: Dave arrives at an airport which has twelve gates arranged in a straight line with exactly feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks feet or less to the new gate be a fraction , where and are relatively prime positive integers. Find .
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Problem 5: Positive numbers , and satisfy and . Find .
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Problem 6: Find the smallest positive integer with the property that the polynomial can be written as a product of two nonconstant polynomials with integer coefficients.
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Problem 7: Let , where , and are real. There exists a complex number such that the three roots of are , and , where . Find .
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Problem 8: Let be the number of ordered pairs of nonempty sets and that have the following properties:
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Problem 9: Let be a regular hexagon. Let , and be the midpoints of sides , and , respectively. The segments , and bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of be expressed as a fraction where and are relatively prime positive integers. Find .
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Problem 10: Find the number of second-degree polynomials with integer coefficients and integer zeros for which .
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Problem 11: Define a -grid to be a matrix which satisfies the following two properties:
Find the number of distinct T-grids.
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Problem 12: Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is . Find the minimum possible value of their common perimeter.
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Problem 13: The cards in a deck are numbered . Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked. The two persons with lower numbered cards form a team, and the two persons with higher numbered cards form another team. Let be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards and , and Dylan picks the other of these two cards. The minimum value of for which can be written as , where and are relatively prime positive integers. Find .
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Problem 14: In right triangle with the right angle at and . Point on has the properties that and . The ratio can be represented 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: In triangle , and . Points and lie on with and . Points and lie on with and . Let be the other point of intersection of the circumcircles of and . Ray meets at . The ratio can be written in the form , where and are relatively prime positive integers. Find .
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The problems on this page are the property of the MAA's American Mathematics Competitions