Positively non-positive

Algebra Level 5

For how many positive integers $N$ between 3 and 1000 (inclusive) is the following statement true?

If $\{ a_i \} _{i=1}^N$ is a set of $N$ (not necessarily distinct) real numbers such that

$a_1 + a_2 + \cdots + a_N = 0,$

then we must (always) have

$a_1 a_2 + a_2 a_3 + \ldots + a_{N-1} a_N + a_N a_1 \leq 0.$

Clarification: The cases of $N=1, N=2$ are removed to avoid ambiguity in the second summation.