This is a three point problem. Consider the game MarkOut played on a 5 by 5 grid of squares. On your turn, you may mark out any number of adjacent squares on any row or column. So, for example, you might mark out the middle three squares on the top row. Your opponent might then mark out the entire first column. You might then mark out the bottom right square. No square can be marked out twice, and you and your opponent alternate moves. The winner is the person who marks out the last square.
This is a game that the first player can always win.
Solution
I believe Jeffery James gave the best explanation:
The first player's winning move is to mark out the center square. Since the board is symmetrical, the first player will then be able to mirror his opponent's moves.
For example, the first player marks the middle square, and the second player marks out the entire bottom row. To mirror this, the first player then marks out the entire top row. Thanks to symmetry, the first player will always have a mirroring move, and thus he will always win.
Jeffery earned four points for his answer, while Jared Latiolais earned four points for a correct answer plus the first correct answer bonus. Other correct answers were received from Mark Goadrich, Chris Chappa, and Paul Ottoway.