Prove that $\frac{a}{b^2 + c}+\frac{b}{c^2 + a}+\frac{c}{a^2 + b} \ge \frac{3}{2}$ when $a+b+c=3$ I've tried solving Pham Kim Hung's famous inequality problem which he posted on aops in April 2007 for 2 days now. This is the problem: Show that $$\frac{a}{b^2 + c}+\frac{b}{c^2 + a}+\frac{c}{a^2 + b} \ge \frac{3}{2} \tag{*}$$
when $a, b, c > 0$ and $a+b+c=3$.
I've tried with many theorems and tricks (like AM-GM, C-S, etc.) but haven't really figured out a way.
The closest I could get was by solving a similar problem like this.
So, if you know the proof of (*) please share here. I'd be much thankful to you.

$\sum_{cyc}\frac{a}{b+c^2}=\sum_{cyc}\frac{a^4}{a^3b+a^3c^2}\geq\frac{(a^2+b^2+c^2)^2}{\frac{1}{3}(a^2+b^2+c^2)^2+\frac{1}{3}(a^2+b^2+c^2)^2}=\frac{3}{2}$
(For numerator we use Titu's Lemma and simplify denominator :) )

Thanks to arqady
For more information vist https://artofproblemsolving.com/community/c6h401290p2234611

