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Algebra Level 5

\[\large\sum_{k=1}^{n-2}x_kx_{k+1}+x_nx_{n-3}>\sum_{k=1}^{n}x_k^2\]

Find the smallest positive integer \(n\geq4\) such that the above inequality holds for some real numbers \(x_1,\ldots,x_n\).


I have posted variants of this problem before, but I'm still waiting for a solution.
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