2024 IMO Shortlist C5

Let $N$ be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers $1,2,\ldots,N$ on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers $(k,n)$, where $k\ge 0$ and $n$ is one of the integers on the board, and then erasing every integer $s$ on the board such that $2^k\mid n-s$. The game continues until the board is empty. The player who erases the last integer on the board loses.

Determine all values of $N$ for which Geoff can ensure that he wins, no matter how Ceri plays.

设 $N$ 为正整数。Geoff 和 Ceri 玩一个游戏：开始时把数字 $1,2,\ldots,N$ 写在黑板上。两人轮流操作，Geoff 先手。每次操作选择一对整数 $(k,n)$，其中 $k\ge 0$，且 $n$ 是黑板上现有的一个整数；然后擦去黑板上所有满足 $2^k\mid n-s$ 的整数 $s$。游戏持续到黑板为空为止。擦去黑板上最后一个整数的玩家输。

确定所有 $N$，使得无论 Ceri 如何应对，Geoff 都能保证获胜。