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Putnam 2008 A2

Source:

December 8, 2008
Putnamlinear algebramatrixcollege contests

Problem Statement

Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008×2008 2008\times 2008 array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?