Question 6
Let $n\ge3$ be an integer and let $p$ be a prime such that $p>6^{n-1}-2^n+1$. Let $S$ consist of $n$ positive integers with pairwise distinct residues modulo $p$. Prove that there is a positive integer $c$ such that exactly two ordered triples $(x,y,z)\in S^3$, with pairwise distinct components, satisfy
$$x-y+z\equiv c\pmod p.$$
$$x-y+z\equiv c\pmod p.$$