Nemotron-IMO-Bench 2026 C42

Real numbers $x_1,...,x_n$ are attached to the $n\geq 2$ vertices of a tree (one number per vertex). For each edge of the tree consider the product of the numbers associated to its endpoints. Prove that the sum $S$ of all products over the edges of the tree satisfies $$S\leq \frac{\sqrt{n-1}}{2}\cdot \left(x_1^2+x_2^2+\dots+x_n^2\right).$$