Nemotron-IMO-Bench 2026 A16

Prove that for all positive real numbers $a,b,c$ $$\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}+\frac{45}{a+b+c}\geq \frac{7}{a}+\frac{7}{b}+\frac{7}{c}.$$