Problem

Source: Stars of Mathematics 2016, Seniors, Problem 2

Tags: inequalities



Let $ n $ be a positive integer and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $ a_1^2+a_2^2+\cdots +a_n^2=1. $ Show that $$ \sum_{1\le ij\le n} a_ia_j<2\sqrt n. $$ Russian math competition