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0543 inequalities 5th edition Round 4 p3

Source:

May 6, 2021
inequalitiesalgebra5th edition

Problem Statement

Let a1,...,ana_1,..., a_n be positive reals and let x1,...,xnx_1, ... , x_n be real numbers such that a1x1+...+anxn=0a_1x_1 +...+ a_nx_n = 0. Prove that ∑1≤i<j≤nxixj∣ai−aj∣≤0.\sum_{1\le i<j \le n} x_ix_j |a_i - a_j | \le 0. When does the equality take place?
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