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( \sum y_k \sqrt{p_k} )^2 <= \sum k/ n p_k

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2011 3.4 C2

September 26, 2021
algebrainequalities

Problem Statement

Let p1,p2,...,pnp_1, p_2, ..., p_n be positive real numbers, such that p1+p2+...+pn=1p_1 + p_2 +... + p_n = 1. Let x∈[0,1]x \in [0,1] and let y1,y2,...,yny_1, y_2, ..., y_n be such that y12+y22+...+yn2=xy^2_1 + y^2_2 +...+ y^2_n= x. Prove that (∑nx≤k≤nykpk)2≤∑k=1nknpk\left( \sum_{nx\le k \le n }y_k \sqrt{p_k} \right)^2 \le \sum_{k=1}^{n}\frac{k}{n} p_k
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