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Let $x_1, x_2, \ldots, x_n$ be $n \geq 2$ real numbers with $x_1+\cdots+x_n \geq 0$ and $x_1^2+\cdots+x_n^2=1$ If $\max \left\{x_1, \ldots, x_n\right\}=M$, prove that
$M \geq \frac{1}{\sqrt{n(n-1)}} .$
Decide if equality is possible.
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