2021 IMO Shortlist C3

A thimblerigger has 2021 thimbles numbered from 1 through 2021. The thimbles are arranged in a circle in arbitrary order. The thimblerigger performs a sequence of 2021 moves; in the $k^{\text {th }}$ move, he swaps the positions of the two thimbles adjacent to thimble $k$. Prove that there exists a value of $k$ such that, in the $k^{\text {th }}$ move, the thimblerigger swaps some thimbles $a$ and $b$ such that $a<k<b$.

顶针装配工有 2021 个顶针，编号从 1 到 2021。顶针以任意顺序排列在圆圈中。顶针操纵者执行一系列 2021 个动作；在$k^{\text {th }}$移动中，他交换了与顶针$k$相邻的两个顶针的位置。证明存在 $k$ 的值，使得在 $k^{\text {th }}$ 移动中，顶针装配者交换了一些顶针 $a$ 和 $b$，使得 $a<k<b$。