2022 APMO 第 4 题

Let $n$ and $k$ be positive integers. Cathy is playing the following game. There are $n$ marbles and $k$ boxes, with the marbles labelled 1 to $n$. Initially, all marbles are placed inside one box. Each turn, Cathy chooses a box and then moves the marbles with the smallest label, say $i$, to either any empty box or the box containing marble $i+1$. Cathy wins if at any point there is a box containing only marble $n$.

Determine all pairs of integers $(n, k)$ such that Cathy can win this game.

令 $n$ 和 $k$ 为正整数。凯茜正在玩以下游戏。有 $n$ 个弹珠和 $k$ 个盒子，弹珠标记为 1 到 $n$。最初，所有弹珠都放在一个盒子里。每回合，Cathy 选择一个盒子，然后将带有最小标签的弹珠(例如 $i$)移动到任何空盒子或包含弹珠 $i+1$ 的盒子。如果在任何时候有一个盒子只包含弹珠 $n$，Cathy 就会获胜。

确定所有整数对 $(n, k)$，使得 Cathy 能够赢得这场游戏。