MathNet 2018-part-04 — 181 problems (EN–VI) Question 97

Determine the positive integers $n \ge 3$ such that, for every integer $m \ge 0$, there exist integers $a_1, a_2, \dots, a_n$ such that $a_1 + a_2 + \dots + a_n = 0$ and $a_1a_2 + a_2a_3 + \dots + a_{n-1}a_n + a_na_1 = -m$.