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.

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