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Game in grid

Source: Pre-VMO 2012 - Round 2 - Problem 4

December 25, 2011
symmetryinductioncombinatorics proposedcombinatorics

Problem Statement

Two people A and B play a game in the m×nm \times n grid (m,n∈N∗m,n \in \mathbb{N^*}). Each person respectively (A plays first) draw a segment between two point of the grid such that this segment doesn't contain any point (except the 2 ends) and also the segment (except the 2 ends) doesn't intersect with any other segments. The last person who can't draw is the loser. Which one (of A and B) have the winning tactics?
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