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China Team Selection Test 2013 TST 3 Day 2 Q2

Source: Nanjing high School , Jiangsu 25 Mar 2013

March 25, 2013
inequalities proposedinequalitiesChina TST

Problem Statement

Let k≥2k\ge 2 be an integer and let a1,a2,⋯ ,an,b1,b2,⋯ ,bna_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\equal{}1}^{n} a_i^2\right)+\left(\frac{1}{n} \sum_{i\equal{}1}^{n} b_i\right)^2\ge\prod_{i=1}^{n}(a_i^{2}+b_i^{2})^{\frac{1}{n}}.
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