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If player B plays correctly, then player A cannot win

Source: Austrian Mathematical Olympiad 1999, Part 2, D2, P3

June 28, 2011
Online Math Opensymmetrycombinatorics proposedcombinatorics

Problem Statement

Two players AA and BB play the following game. An even number of cells are placed on a circle. AA begins and AA and BB play alternately, where each move consists of choosing a free cell and writing either OO or MM in it. The player after whose move the word OMOOMO (OMO = Osterreichische Mathematik Olympiade) occurs for the first time in three successive cells wins the game. If no such word occurs, then the game is a draw. Prove that if player BB plays correctly, then player AA cannot win.