2017 IMO Shortlist A5

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ Shiny if for each permutation $y_{1}, y_{2}, \ldots, y_{n}$ of these numbers we have $$\sum_{i=1}^{n-1} y_{i} y_{i+1}=y_{1} y_{2}+y_{2} y_{3}+y_{3} y_{4}+\cdots+y_{n-1} y_{n} \geq-1$$ Find the largest constant $K=K(n)$ such that $$\sum_{1 \leq i<j \leq n} x_{i} x_{j} \geq K$$ holds for every Shiny $n$-tuple $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$.

给出一个整数 $n \geq 3$。如果对于这些数字的每个排列 $y_{1}, y_{2}, \ldots, y_{n}$ 我们有 $$ \sum_{i=1}^{n-1} y_{i} y_{i+1}=y_{1} ，我们称 $n$ 实数元组 $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ 为闪亮y_{2}+y_{2} y_{3}+y_{3} y_{4}+\cdots+y_{n-1} y_{n} \geq-1 $$ 找到最大常数 $K=K(n)$，使得 $$\sum_{1 \leq i<j \leq n} x_{i} x_{j} \geq K$$ 对每个闪亮 $n$ 元组都成立$\left(x_{1}, x_{2}, \ldots, x_{n}\right)$。