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China Team Selection Test 2013 TST 2 Day 2 Q3

Source: Nanjing high School , Jiangsu 19 Mar 2013

March 19, 2013

inequalitiesChina TSTalgebra

Problem Statement

Let

n>1

be an integer and let

a_0,a_1,\ldots,a_n

be non-negative real numbers. Definite S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i for

k=0,1,\ldots,n

. Prove that\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.