Question 25
Let $n\ge3$ be an integer and let $p$ be a prime such that $p\gt6^{n-1}-2^n+1$. Let $S$ be a set of $n$ positive integers with pairwise distinct residues modulo $p$. Prove that there is a positive integer $c$ for which exactly two ordered triples $(x,y,z)\in S^3$ with pairwise distinct entries satisfy $p\mid x-y+z-c$.