CR-MO 2011 Problem 4

Let $p_1, p_2, ..., p_n$ be positive real numbers, such that $p_1 + p_2 +... + p_n = 1$ . Let $x \in [0,1]$ and let $y_1, y_2, ..., y_n$ be such that $y^2_1 + y^2_2 +...+ y^2_n= x$ . Prove that $$\left( \sum_{nx\le k \le n }y_k \sqrt{p_k} \right)^2 \le \sum_{k=1}^{n}\frac{k}{n} p_k$$