BALKAN-SL 2020 N5

Consider an integer $n\geq 2$ and an odd prime $p$ . Let $U$ be the set of all positive integers $($ strictly $)$ less than $p^n$ that are not divisible by $p$ , and let $N$ be the number of elements of $U$ . Does there exist permutation $a_1,a_2,\cdots a_N$ of the numbers in $U$ such that the sum $\sum_{k=1}^N a_ka_{k+1}$ ,where $a_{N+1}=a_1$ , be divisible by $p^{n-1}$ but not by $p^n$ ? $Alexander \ Ivanov Bulgaria$