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River Li

Let $3 \le n\le 19$ be an integer. Let $x_1, x_2, \cdots, x_n \ge 0$ and $x_1^2 + x_2^2 + \cdots + x_n^2 = 1$. Prove that $$\sqrt{1-x_1x_2}+\sqrt{1-x_2x_3}+\cdots+\sqrt{1-x_{n-1}x_n}+\sqrt{1-x_nx_1}\ge \sqrt{n(n-1)}.$$ (Remark: When $n\ge 20$, the inequality does not hold.)

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