Difference between revisions of "2005 AIME II Problems/Problem 6"
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== Problem == | == Problem == | ||
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The cards in a stack of <math> 2n </math> cards are numbered consecutively from 1 through <math> 2n </math> from top to bottom. The top <math> n </math> cards are removed, kept in order, and form pile <math> A. </math> The remaining cards form pile <math> B. </math> The cards are then restacked by taking cards alternately from the tops of pile <math> B </math> and <math> A, </math> respectively. In this process, card number <math> (n+1) </math> becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles <math> A </math> and <math> B </math> 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 named magical. Find the number of cards in the magical stack in which card number 131 retains its original position. | The cards in a stack of <math> 2n </math> cards are numbered consecutively from 1 through <math> 2n </math> from top to bottom. The top <math> n </math> cards are removed, kept in order, and form pile <math> A. </math> The remaining cards form pile <math> B. </math> The cards are then restacked by taking cards alternately from the tops of pile <math> B </math> and <math> A, </math> respectively. In this process, card number <math> (n+1) </math> becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles <math> A </math> and <math> B </math> 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 named magical. Find the number of cards in the magical stack in which card number 131 retains its original position. | ||
== Solution == | == Solution == | ||
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== See Also == | == See Also == | ||
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*[[2005 AIME II Problems]] | *[[2005 AIME II Problems]] | ||
Revision as of 21:46, 5 September 2006
Problem
The cards in a stack of 
cards are numbered consecutively from 1 through from top to bottom. The top 
cards are removed, kept in order, and form pile 
The remaining cards form pile 
The cards are then restacked by taking cards alternately from the tops of pile 
and 
respectively. In this process, card number 
becomes the bottom card of the new stack, card number 1 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 named magical. Find the number of cards in the magical stack in which card number 131 retains its original position.
Solution
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