BALTIC-WAY 1999 Problem 3

Determine all positive integers $n \geqslant 3$ such that the inequality

$$a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{n-1} a_{n}+a_{n} a_{1} \leqslant 0$$

holds for all real numbers $a_{1}, a_{2}, \ldots, a_{n}$ which satisfy $a_{1}+\cdots+a_{n}=0$.