Question 6
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
$$ z_{i+j}^{2}\geq x_{i}y_{j}\qquad\text{for all }1\leq i,j\leq n. $$
Let $ M = \max\{z_2, \ldots, z_{2n}\} $. Prove that
$$ \left(\frac{M+z_{2}+\cdots+z_{2n}}{2n}\right)^{2}\geq\left(\frac{x_{1}+\cdots+x_{n}}{n}\right)\left(\frac{y_{1}+\cdots+y_{n}}{n}\right). $$
$$ z_{i+j}^{2}\geq x_{i}y_{j}\qquad\text{for all }1\leq i,j\leq n. $$
Let $ M = \max\{z_2, \ldots, z_{2n}\} $. Prove that
$$ \left(\frac{M+z_{2}+\cdots+z_{2n}}{2n}\right)^{2}\geq\left(\frac{x_{1}+\cdots+x_{n}}{n}\right)\left(\frac{y_{1}+\cdots+y_{n}}{n}\right). $$