Problem

Source: Kosovo Mathematical Olympiad 2022, Grade 10, Problem 4

Tags: inequalities, algebra



Let $a,b$ and $c$ be positive real numbers such that $a+b+c+3abc\geq (ab)^2+(bc)^2+(ca)^2+3$. Show that the following inequality hold, $$\frac{a^3+b^3+c^3}{3}\geq\frac{abc+2021}{2022}.$$