, 2021

Alice and Bob take it in turns to write numbers on a blackboard. Alice starts by writing an integer $a$ between $-100$ and $100$ inclusive on the board. On each of Bob's turns he writes twice the number Alice wrote last. On each of Alice's subsequent turns she writes the number $45$ less than the number Bob wrote last. At some point, the number $a$ is written on the board for a second time. Find the possible values of $a$.