Question 112
An odd integer $a \gt 1$ is given. Initially, Basil chooses an even positive integer $b$ such that $b \lt a$ and tells it to Pete. Basil then writes down three integers on a blackboard. After that, Pete makes a sequence of moves. By a move, Pete can either add $a$ to one of the numbers on the blackboard, add $b$ to the second number, and subtract $a+b+1$ from the third number, or, conversely, subtract $a$ from one number on the blackboard, subtract $b$ from the second number, and add $a+b+1$ to the third one. At each move, Pete can independently choose which number on the blackboard is the first, the second, and the third. Pete wins if, after some moves, all three numbers on the blackboard are zero. For which $a$ Basil cannot prevent Pete's win?