IRL_ABooklet_2023

Question

Suppose a_1, a_2, , a_n are positive numbers. Prove that the square root of the arithmetic mean of the numbers a_i a_j, 1 i < j n doesn't exceed the arithmetic mean of a_k, k = 1, 2, , n, with equality iff a_1 = a_2 = = a_n.

Accepted Answer

The claim is that _1 i < j n a_i a_jn2 ( _k=1^n a_kn )^2 . Equivalently, 2n _1 i < j n a_i a_j (n-1) ( _k=1^n a_k^2 + 2 _1 i < j n a_i a_j ), or 2 _1 i < j n a_i a_j (n-1) _k=1^n a_k^2, i.e., ( _k=1^n a_k )^2 n _k=1^n a_k^2, which is true by the Cauchy-Schwarz inequality. There is equality iff a_1 = a_2 = = a_n.