AoPS - Problem

Solved by Leaf_Old. It's well known that $$ \begin{aligned} &9(a+b)(b+c)(c+a)\ge8(a+b+c)(ab+bc+ca)\\ \Longrightarrow\;\;&\frac{3}{ab+bc+ca}\ge\frac13\sum_{cyc}\frac4{(a+b)(b+c)}. \end{aligned} $$So by applying it to LHS we have $$ \begin{aligned} \mathrm{LHS}&\ge n\sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\left(\frac13\sum_{cyc}\frac4{(a_i+a_j)(a_j+a_k)}\right)\\ &=n\sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\sum_{cyc}\frac4{(a_i+a_j)(a_j+a_k)}\\ &=n\sum_{i=1}^n\left(\sum_{j=1}^n\frac2{a_i+a_j}\right)\left(\sum_{k=1}^n\frac2{a_i+a_k}\right)\\ &=n\sum_{i=1}^n\left(\sum_{j=1}^n\frac2{a_i+a_j}\right)^2. \end{aligned} $$Let $\sum\limits_{j=1}^n\dfrac2{a_i+a_j}=x_i$, by Cauchy-Schwarz we have $$\mathrm{LHS}\ge\left(\sum_{i=1}^n x_i\right)^2=\mathrm{RHS}.$$