Two players place coins of identical size (say quarters) on a round table. Each player has to place exactly one coin on the table without overlap with the coins already on the table. The first player who cannot put a coin on the table loses.
Prove that the first player has a winning strategy.
The first player puts a coin at the center of the table and after that, whatever position of the table the second player puts a coin, the first player keeps his coin at a position which is the reflection of the previous coin through the center coin i.e. rotated through $180$ degrees. Or in other words, as fleablood puts it; the position is "collinear to the center and the previous coin at an equal distance from the center as the previous coin".