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double sum (t_i-t_j)^2x_ix_j <= 0 if sum x_i = 0

Source: Netherlands - Dutch NMO 1975, year 1975-76 (also named as 1975 -2) p3

January 27, 2023

inequalitiesSumalgebra

Problem Statement

Given are the real numbers

x_1,x_2,...,x_n

and

t_1,t_2,...,t_n

for which holds:

\sum_{i=1}^n x_i = 0

. Prove that

\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.