RO-NMO 2021 Problem 82

Prove that for all positive real numbers $a,b,c$ the following inequality holds: $$(a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\ge\frac{2(a^2+b^2+c^2)}{ab+bc+ca}+7$$ and determine all cases of equality.

Lucian Petrescu