Problem

Source: 2022 Junior Macedonian Mathematical Olympiad P2

Tags: inequalities, algebra, rational function, function



Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=3$. Prove the inequality $$\frac{a^3}{a^2+1}+\frac{b^3}{b^2+1}+\frac{c^3}{c^2+1} \geq \frac{3}{2}.$$ Proposed by Anastasija Trajanova