MathDB
A tree with numbers in vertices and edges

Source: All-Russian Olympiad, 2003, grade 10, day 1, no. 3

September 1, 2011
inductioninequalitiescombinatorics

Problem Statement

A tree with n≥2n\geq 2 vertices is given. (A tree is a connected graph without cycles.) The vertices of the tree have real numbers x1,x2,…,xnx_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 SS be a sum of number across all edges. Prove that n−1(x12+x22+⋯+xn2)≥2S.\sqrt{n-1}\left(x_1^2+x_2^2+\dots+x_n^2\right)\geq 2S.
(Author: V. Dolnikov)
Back to ProblemsView on AoPS