DE-BUNDESWETTBEWERB 2005 Problem 5

Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts. Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$ 's stone is lying on at that moment.