Question 37
Let $t$, $a$ and $b$ be positive integers. We call a $(t ; a, b)$-game the following game with two players: the first player subtracts $a$ or $b$ from $t$, then the second player subtracts $a$ or $b$ from the number obtained by the first player, then again the first player subtracts $a$ or $b$ from the number obtained by the second player and so on. The player who obtains first a negative number loses the game. Prove that there exist infinitely many $t$ such that the first player has a winning strategy for any $(t ; a, b)$-game with $a+b=2005$.