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Game about divisibility

Source: Middle European Mathematical Olympiad 2013 T-8

May 17, 2014
number theory proposednumber theory

Problem Statement

The expression ±□±□±□±□±□±□ \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box is written on the blackboard. Two players, A A and B B , play a game, taking turns. Player A A takes the first turn. In each turn, the player on turn replaces a symbol □ \Box by a positive integer. After all the symbols □\Box are replace, player AA replaces each of the signs ±\pm by either + or -, independently of each other. Player A A wins if the value of the expression on the blackboard is not divisible by any of the numbers 11,12,⋯ ,18 11, 12, \cdots, 18 . Otherwise, player B B wins. Determine which player has a winning strategy.
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