Let be an even positive integer. Show that there is a permutation of such that for every , the number is one of the numbers , , , . Hereby, we use the cyclic subscript convention, so that means .
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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 $x_{1}$.
Source: Međunarodna matematička olimpijada, shortlist 2002
Školjka 
be an even positive integer. Show that there is a permutation 
of 
such that for every 
, the number 
is one of the numbers 
, 
, 
, 
. Hereby, we use the cyclic subscript convention, so that 
means 
.