2005 AIME II Problems/Problem 6
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.