Question 90
Let $n \geq 1$ be a natural number and let $x_{1}, \ldots, x_{n}$ be strictly positive real numbers. Show that one can choose $a_{1}, \ldots, a_{n} \in \{-1,1\}$ such that
$$
\sum_{i=1}^{n} a_{i} x_{i}^{2} \geq \left(\sum_{i=1}^{n} a_{i} x_{i}\right)^{2}
$$
$$
\sum_{i=1}^{n} a_{i} x_{i}^{2} \geq \left(\sum_{i=1}^{n} a_{i} x_{i}\right)^{2}
$$