Seemingly Innocent Inequality

Algebra Level 5

k=1nxk2k=1n1xk2\sum_{k=1}^{n} x_k^2 \le \sum_{k=1}^n \dfrac{1}{x_k^2}

Given that x1,x2,,xnx_1,x_2, \ldots,x_n are positive reals whose sum is nn, find the largest integer nn such that the inequality above always holds true.

If you think all positive integers nn make the inequality hold true, enter 0 as your answer.

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