2022 IMO Shortlist C4

Let $n>3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbours on the right and left. Determine all initial distributions of coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

设$n>3$为正整数。假设$n$个孩子排成一圈，他们之间分配了$n$个硬币(有些孩子可能没有硬币)。每走一步，拥有至少 2 个硬币的孩子可以给左右两侧的每个近邻各 1 个硬币。确定硬币的所有初始分布，在有限步数之后，每个孩子可能恰好拥有一枚硬币。