2002 IMO Shortlist S21

Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1,\,2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$ , the number $x_{i+1}$ is one of the numbers $2x_{i}$ , $2x_{i}-1$ , $2x_{i}-n$ , $2x_{i}-n-1$ . Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$ .

令 $n$ 为偶正整数。证明存在 $\left(1,\,2,\,\ldots,n\right)$ 的排列 $\left(x_{1},x_{2},\ldots,x_{n}\right)$ ，使得对于每个 $i\in\left\{1,\ 2,\ ...,\ n\right\}$ ，数字 $x_{i+1}$ 是数字 $2x_{i}$ 之一， $2x_{i}-1$ 、 $2x_{i}-n$ 、 $2x_{i}-n-1$ 。因此，我们使用循环下标约定，因此 $x_{n+1}$ 表示 $x_{1}$ 。