Consider the following problem from Problems from the Book proposed by Gabriel Dospinescu
Let $a_1, a_2, \dots, a_n$ be positive real numbers and let $S = a_1 + a_2 + \dots + a_n$ be their sum. Prove that $$ \frac{1}{n} \sum_{i = 1}^n \frac{1}{a_i} + \frac{n(n-2)}{S} \geq \sum_{i \neq j} \frac{1}{S + a_i - a_j}$$
I've been trying to solve this using standard methods such as fudging and Cauchy-Schwarz without any luck (the inequality would become too weak). Any suggestions or solutions?