Question 78
An integer $n \geqslant 3$ is given. We call an $n$-tuple of real numbers $(x_{1}, x_{2}, \ldots, x_{n})$ Shiny if for each permutation $y_{1}, y_{2}, \ldots, y_{n}$ of these numbers we have
$$
\sum_{i=1}^{n-1} y_{i} y_{i+1} = y_{1} y_{2} + y_{2} y_{3} + y_{3} y_{4} + \cdots + y_{n-1} y_{n} \geqslant -1.
$$
Find the largest constant $K = K(n)$ such that
$$
\sum_{1 \leqslant i \lt j \leqslant n} x_{i} x_{j} \geqslant K
$$
holds for every Shiny $n$-tuple $(x_{1}, x_{2}, \ldots, x_{n})$.
$$
\sum_{i=1}^{n-1} y_{i} y_{i+1} = y_{1} y_{2} + y_{2} y_{3} + y_{3} y_{4} + \cdots + y_{n-1} y_{n} \geqslant -1.
$$
Find the largest constant $K = K(n)$ such that
$$
\sum_{1 \leqslant i \lt j \leqslant n} x_{i} x_{j} \geqslant K
$$
holds for every Shiny $n$-tuple $(x_{1}, x_{2}, \ldots, x_{n})$.