RU-ALLRUSSIAN 2003 Problem 103

A tree with $n\geq 2$ vertices is given. (A tree is a connected graph without cycles.) The vertices of the tree have real numbers $x_1,x_2,\dots,x_n$ associated with them. Each edge is associated with the product of the two numbers corresponding to the vertices it connects. Let $S$ be a sum of number across all edges. Prove that $$\sqrt{n-1}\left(x_1^2+x_2^2+\dots+x_n^2\right)\geq 2S.$$

(Author: V. Dolnikov)