Problem

Source: Serbia JBMO TST 2021

Tags: algebra, inequalities



Prove that for positive real numbers $a, b, c$ the following inequality holds: \begin{align*} \frac{a}{9bc+1}+\frac{b}{9ca+1}+\frac{c}{9ab+1}\geq \frac{a+b+c}{1+(a+b+c)^2} \end{align*}When does equality occur?