Question 58
Let $n$ be a positive integer. Prove that the inequality
$$
n \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} \frac{3}{a_j a_k + a_k a_i + a_i a_j} \ge \left( \sum_{j=1}^{n} \sum_{k=1}^{n} \frac{2}{a_j + a_k} \right)^2
$$
holds for any positive real numbers $a_1, a_2, \dots, a_n$.
$$
n \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} \frac{3}{a_j a_k + a_k a_i + a_i a_j} \ge \left( \sum_{j=1}^{n} \sum_{k=1}^{n} \frac{2}{a_j + a_k} \right)^2
$$
holds for any positive real numbers $a_1, a_2, \dots, a_n$.