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2-player game with a square, drawing segments of given segment

Source: 2011 Argentina OMA L3 p6

June 17, 2020
geometrycombinatoricsgamesegmentssquaregame strategywinning strategy

Problem Statement

We have a square of side 11 and a number \ell such that 0<<20 <\ell <\sqrt2. Two players AA and BB, in turn, draw in the square an open segment (without its two ends) of length \ell , starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.