Question 111
Let $n \geqslant 2$ and let $p$ be an odd prime number. Let $U$ be the set of positive integers less than or equal to $p^{n}$ and coprime to $p$, and let $N=|U|$. Show that there exists a permutation $a_{1}, \ldots, a_{N}$ of the elements of $U$ such that $\sum_{k=1}^{N} a_{k} a_{k+1}$ (with $a_{N+1}=a_{1}$) is divisible by $p^{n-1}$ but not by $p^{n}$.