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Positively non-positive

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

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

a1+a2++aN=0, a_1 + a_2 + \cdots + a_N = 0,

then we must (always) have

a1a2+a2a3++aN1aN+aNa10. 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=2N=1, N=2 are removed to avoid ambiguity in the second summation.


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