Question 121
Let $n \ge 3$ be a positive integer and let $p$ be a prime satisfying $p \gt 6^{n-1} - 2^n + 1$. Let $S$ be a set of $n$ positive integers, any two of which have different remainders upon division by $p$. Prove that there exists a positive integer $c$ such that there are exactly two ordered triples $(x, y, z) \in S^3$ with $x, y, z$ pairwise distinct and $x - y + z - c$ divisible by $p$.