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Consider the following problem from Problems from the Book proposed by Gabriel Dospinescu

Let $a_1, a_2, ..., a_n$ be positive real numbers and let $S = a_1 + a_2 + ... + a_n$ 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-Schwartz without any luck (the inequality would become too weak). Any suggestions or solutions?

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Hints:
1. Use Karamata’s inequality for the convex function $t\mapsto \dfrac1t$. You will then need to show $$(\underbrace{na_1, na_2, \dots na_n}_{n \text{ terms}}, \underbrace{S, S, \dots, S}_{n(n-2) \text{ times}}) \succ (\underbrace{S+a_1-a_2, S+a_1-a_3, \dots S+a_n-a_{n-1}}_{n(n-1) \text{ terms}})$$

  1. The majorisation can be established by noting WLOG $a_k$ can be ordered non-decreasing say...
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