Let $a_1,..., a_n$ be positive reals and let $x_1, ... , x_n$ be real numbers such that $a_1x_1 +...+ a_nx_n = 0$. Prove that $$\sum_{1\le i<j \le n} x_ix_j |a_i - a_j | \le 0.$$When does the equality take place?
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Tags: inequalities, algebra, 5th edition
Let $a_1,..., a_n$ be positive reals and let $x_1, ... , x_n$ be real numbers such that $a_1x_1 +...+ a_nx_n = 0$. Prove that $$\sum_{1\le i<j \le n} x_ix_j |a_i - a_j | \le 0.$$When does the equality take place?