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A game in a 1xm board

Source: Turkish TST 2012 Problem 6

March 26, 2012
LaTeXcombinatorics proposedcombinatorics

Problem Statement

Two players AA and BB play a game on a 1×m1\times m board, using 20122012 pieces numbered from 11 to 2012.2012. At each turn, AA chooses a piece and BB places it to an empty place. After kk turns, if all pieces are placed on the board increasingly, then BB wins, otherwise AA wins. For which values of (m,k)(m,k) pairs can BB guarantee to win?
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