Question 159
Let $n \geqslant 2$ and let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers such that $x_{1}+x_{2}+\cdots+x_{n}=0$ and $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$.
Prove that there exists $i$ such that $x_{i} \geqslant \frac{1}{\sqrt{n(n-1)}}$.
Prove that there exists $i$ such that $x_{i} \geqslant \frac{1}{\sqrt{n(n-1)}}$.