ISL 2003 A6

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$ , $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$ .

Let $M=\max\{z_2,\ldots,z_{2n}\}$ . Prove that $$\left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right).$$

Proposed by Reid Barton, USA

Attachments: