2023 IMO Shortlist C1

Let $m$ and $n$ be positive integers greater than 1 . In each unit square of an $m \times n$ grid lies a coin with its tail-side up. A move consists of the following steps: 1. select a $2 \times 2$ square in the grid; 2. flip the coins in the top-left and bottom-right unit squares; 3. flip the coin in either the top-right or bottom-left unit square. Determine all pairs $(m, n)$ for which it is possible that every coin shows head-side up after a finite number of moves. (Thailand)

令 $m$ 和 $n$ 为大于 1 的正整数。在 $m \times n$ 网格的每个单位正方形中都有一枚硬币，硬币的尾部朝上。移动包括以下步骤： 1. 在网格中选择一个 $2 \times 2$ 的方格； 2. 翻转左上角和右下单位方块中的硬币； 3. 将硬币投掷到右上角或左下角的单位方块中。确定所有对 $(m, n)$，其中每枚硬币在有限次数的移动后可能会正面朝上。 (泰国)