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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 \qquad for all 1\le i,j \leq n.


Let M=\max\{z_2,\ldots,z_{2n}\}. Prove that

\biggl(\frac{M+z_2+\cdots+z_{2n}}{2n}\biggr)^2\ge \biggl(\frac{x_1+\cdots+x_n}{n}\biggr)\biggl(\frac{y_1+\cdots+y_n}{n}\biggr...

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Edited by Orl.

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