MathNet 2022-part-03 — 197 problems (EN–VI) Question 153

Let $n \ge 1$. Let $a_1, \dots, a_{2n+2}$ be a sequence of pairwise distinct integers. Prove that $a_{2n+2} - a_1$ is divisible by $2n+1$ if we have $|a_i - a_j| \le n$ whenever $|i-j| \le n$.