USAMO 2020 Problem 6

Let $n \geq 2$ be an integer. Let $x_{1} \geq x_{2} \geq \cdots \geq x_{n}$ and $y_{1} \geq y_{2} \geq \cdots \geq y_{n}$ be $2 n$ real numbers such that $$\begin{aligned} 0 & =x_{1}+x_{2}+\cdots+x_{n}=y_{1}+y_{2}+\cdots+y_{n} \\ \text { and } 1 & =x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\cdots+y_{n}^{2} \end{aligned}$$ Prove that $$\sum_{i=1}^{n}\left(x_{i} y_{i}-x_{i} y_{n+1-i}\right) \geq \frac{2}{\sqrt{n-1}}$$