ISL 2002 C6

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 $x_{1}$ .

Attachments: