Question 40
Let $n \geq 2$ and let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying $x_{1}+x_{2}+\ldots+x_{n} \geq 0$ and $x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=1$. Let $M=\max \{x_{1}, x_{2}, \ldots, x_{n}\}$. Show that
$$
M \geq \frac{1}{\sqrt{n(n-1)}}
$$
When does equality hold in (1)?
$$
M \geq \frac{1}{\sqrt{n(n-1)}}
$$
When does equality hold in (1)?