MathDB

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).

[hide="comment"] Edited by Orl.
Proposed by Reid Barton, USA

Problem Statement