2022 IMO Shortlist C2

The Bank of Oslo issues coins made out of two types of metal: aluminium (denoted A) and copper (denoted $C$ ). Morgane has $n$ aluminium coins, and $n$ copper coins, and arranges her $2 n$ coins in a row in some arbitrary initial order. Given a fixed positive integer $k \leq 2 n$, she repeatedly performs the following operation: identify the largest subsequence containing the $k$-th coin from the left which consists of consecutive coins made of the same metal, and move all coins in that subsequence to the left end of the row. For example, if $n=4$ and $k=4$, the process starting from the configuration $A A C C C A C A$ would be $$A A C C C A C A \rightarrow C C C A A A C A \rightarrow A A A C C C C A \rightarrow C C C C A A A A \rightarrow \cdots$$ Find all pairs $(n, k)$ with $1 \leq k \leq 2 n$ such that for every initial configuration, at some point of the process there will be at most one aluminium coin adjacent to a copper coin. (France)

奥斯陆银行发行由两种金属制成的硬币：铝(表示为 A)和铜(表示为 $C$)。 Morgane 有 $n$ 铝币和 $n$ 铜币，并以某种任意的初始顺序将她的 $2 n$ 硬币排成一排。给定一个固定正整数 $k \leq 2 n$，她重复执行以下操作：找出包含左侧第 $k$ 个硬币(由相同金属制成的连续硬币组成)的最大子序列，并将该子序列中的所有硬币移动到该行的左端。例如，如果 $n=4$ 且 $k=4$，则从配置 $A A C C C A C A$ 开始的过程将为 $$A A C C C C A C A \rightarrow C C C A A A C A \rightarrow A A A C C C C A \rightarrow C C C C A A A A \rightarrow \cdots$$ 查找所有 $(n, k)$ 与 $1 \leq k \leq 2 n$ 的对，这样对于每个初始配置，在该过程的某个时刻，最多会出现一枚铝币与一枚铜币相邻。 (法国)