Question 97
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be real numbers and $c_1, \dots, c_n$ be positive real numbers. Prove that
$$
\left( \sum_{i,j=1}^{n} \frac{a_i a_j}{c_i + c_j} \right) \left( \sum_{i,j=1}^{n} \frac{b_i b_j}{c_i + c_j} \right) \ge \left( \sum_{i,j=1}^{n} \frac{a_i b_j}{c_i + c_j} \right)^2 .
$$
$$
\left( \sum_{i,j=1}^{n} \frac{a_i a_j}{c_i + c_j} \right) \left( \sum_{i,j=1}^{n} \frac{b_i b_j}{c_i + c_j} \right) \ge \left( \sum_{i,j=1}^{n} \frac{a_i b_j}{c_i + c_j} \right)^2 .
$$