Let $k\ge 2$ be an integer and let $a_1 ,a_2 ,\cdots ,a_n,b_1 ,b_2 ,\cdots ,b_n$ be non-negative real numbers. Prove that\[\left(\frac{n}{n-1}\right)^{n-1}\left(\frac{1}{n} \sum_{i=1}^{n} a_i^2\right)+\left(\frac{1}{n} \sum_{i=1}^{n} b_i\right)^2\ge\prod_{i=1}^{n}(a_i^{2}+b_i^{2})^{\frac{1}{n}}.\]
Problem
Source: Nanjing high School , Jiangsu 25 Mar 2013
Tags: inequalities proposed, inequalities, China TST
