Problem Set Workbook
Access the downloadable workbook for 2005 AIME II problems here.
Discussion Forum
Engage in discussion about the 2005 AIME II math contest by visiting Random Math AIME II 2005 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2005 AIME II problems, please refer below:
Problem 1: A game uses a deck of different cards, where is an integer and . The number of possible sets of cards that can be drawn from the deck is times the number of possible sets of cards that can be drawn. Find .
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Problem 2: A hotel packed a breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls, and, once they were wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability that each guest got one roll of each type is , where and are relatively prime positive integers, find .
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Problem 3: An infinite geometric series has sum . A new series, obtained by squaring each term of the original series, has sum times the sum of the original series. The common ratio of the original series is , where and are relatively prime positive integers. Find .
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Problem 4: Find the number of positive integers that are divisors of at least one of , 18^
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Problem 5: Determine the number of ordered pairs of integers such that , and .
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Problem 6: The cards in a stack of cards are numbered consecutively from through from top to bottom. The top cards are removed, kept in order, and form pile . The remaining cards form pile . The cards are now restacked into a single stack by taking cards alternately from the tops of pile and pile , respectively. In this process, card number is the bottom card of the new stack, card number is on top of this card, and so on, until piles and are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is called magical. For example, eight cards form a magical stack because cards number and number retain their original positions. Find the number of cards in the magical stack in which card number retains its original position.
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Problem 7: Let
Find .
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Problem 8: Circles and are externally tangent, and they are both internally tangent to circle . The radii of and are and , respectively, and the centers of the three circles are collinear. A chord of is also a common external tangent of and . Given that the length of the chord is , 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 9: For how many positive integers less than or equal to is
true for all real
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Problem 10: Given that is a regular octahedron, that is the cube whose vertices are the centers of the faces of , and that the ratio of the volume of to that of is , where and are relatively prime positive integers, find .
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Problem 11: Let be a positive integer, and let be a sequence of real numbers such that , and
for . Find .
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Problem 12: Square has center and are on with and between and , and . Given that , where , and are positive integers and is not divisible by the square of any prime, find .
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Problem 13: Let be a polynomial with integer coefficients that satisfies and . Given that the equation has two distinct integer solutions and , find the product .
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Problem 14: In , and . Point is on with . Point is on such that . Given that , where and are relatively prime positive integers, find .
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Problem 15: Let and denote the circles and , respectively. Let be the smallest positive value of for which the line contains the center of a circle that is internally tangent to and externally tangent to . Given that , 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