TH-TMO 2020 Problem 8

For all positive real numbers $a,b,c$ with $a+b+c=3$ , prove the inequality $$\frac{a^6}{c^2+2b^3} + \frac{b^6}{a^2+2c^3} + \frac{c^6}{b^2+2a^3} \geq 1.$$