3. (3 points) Petya and Vasya are playing a game on an initially white $100 \times 100$ grid. Petya goes first and can paint one cell black with his first move. Each subsequent move allows a player to paint black any vertical or horizontal white rectangular strip $1 \times n$ on the grid, where $n$ is a natural number, and it can either match the number of cells just painted by the other player or exceed it by one. The player who cannot make a move loses. Who will win with correct play by both opponents?
Answer: The second player, Vasya, wins.

Solution: He moves symmetrically to his opponent relative to the center of the board. Thus, after each of his moves, the position on the board is symmetric relative to the center, which means that for any move of the opponent, he again has a symmetric response. Therefore, the one who runs out of moves can only be the second, meaning the first wins.