2003 CentroAmerican #1

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Two players $A$ and $B$ take turns playing the following game: There is a pile of $2003$ stones. In his first turn, $A$ selects a divisor of $2003$ and removes this number of stones from the pile. $B$ then chooses a divisor of the number of remaining stones, and removes that number of stones from the new pile, and so on. The player who has to remove the last stone loses. Show that one of the two players has a winning strategy and describe the strategy.



 After A takes the first stone, B wins by taking an odd number of stones on each turn. Since odd numbers only have odd divisors, A will always be forced to leave an even number of stones, thus allowing B to force A to take the last stone which is an odd number.$\square$




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