Question 74
Real numbers $a_1, a_2, \dots, a_n$ ($n \ge 2$) satisfy $-1 \lt a_1, a_2, \dots, a_n \lt 1$ and $\sum_{k=1}^n a_k^2 \ge 1$. Prove
$$
\sum_{i\lt j} \left( \frac{1}{1-a_i^2} + \frac{1}{1-a_j^2} - \frac{1}{1-a_i a_j} \right) \ge \frac{n^2}{2}
$$
$$
\sum_{i\lt j} \left( \frac{1}{1-a_i^2} + \frac{1}{1-a_j^2} - \frac{1}{1-a_i a_j} \right) \ge \frac{n^2}{2}
$$