Question 28
Let $n \ge 3$ be a given positive integer. We call an $n$-tuple of real numbers $(x_1, x_2, \cdots, x_n)$ *shiny* if for every permutation $y_1, y_2, \cdots, y_n$ of $x_1, x_2, \cdots, x_n$ we have
$$
\sum_{i=1}^{n-1} y_i y_{i+1} = y_1 y_2 + y_2 y_3 + y_3 y_4 + \cdots + y_{n-1} y_n \ge -1.
$$
Find the largest number $K = K(n)$ such that for every shiny $n$-tuple $(x_1, x_2, \cdots, x_n)$,
$$
\sum_{1 \le i \lt j \le n} x_i x_j \ge K
$$
holds.
$$
\sum_{i=1}^{n-1} y_i y_{i+1} = y_1 y_2 + y_2 y_3 + y_3 y_4 + \cdots + y_{n-1} y_n \ge -1.
$$
Find the largest number $K = K(n)$ such that for every shiny $n$-tuple $(x_1, x_2, \cdots, x_n)$,
$$
\sum_{1 \le i \lt j \le n} x_i x_j \ge K
$$
holds.