Problem

Source: Kazakhstan Junior National Olympiad , i e 9 grade, problem 5

Tags: inequalities, Kazakhstan 2022, inequalities proposed, algebra



For positive reals $a,b,c$ with $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 3$ prove that $$\frac{a^3}{a^2+b}+\frac{b^3}{b^2+c}+\frac{c^3}{c^2+a}\ge \frac{3}{2}$$