Let n be a positive integer and let $a_1, a_2, ..., a_n, b_1, b_2, ... , b_n, c_2, c_3, ... , c_{2n}$ be $4n - 1$ positive real numbers such that $c^2_{i+j} \ge a_ib_j $, for all $1 \le i, j \le n$. Also let $m = \max_{2 \le i\le 2n} c_i$. Prove that $$\left(\frac{m + c_2 + c_3 +... + c_{2n}}{2n} \right)^2 \ge \left(\frac{a_1+a_2 + ... +a_n}{n}\right)\left(\frac{ b_1 + b_2 + ...+ b_n}{n}\right)$$