Question 57
Consider an integer $n \ge 2$ 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 a permutation $a_1, a_2, \dots, a_N$ of the numbers in $U$ such that the sum $\sum_{k=1}^N a_k a_{k+1}$, where $a_{N+1} = a_1$, be divisible by $p^{n-1}$, but not by $p^n$?
Alexander Ivanov, Bulgaria
Alexander Ivanov, Bulgaria