For positive, real $a$, $b$, $c$ with $a+b+c=1$, show $\sum_{cyc}\frac{a^3}{a^2+b^2}\geq \frac12$

Question

I have some problem solving the following inequality:

Let's assume that $a$, $b$ and $c$ are all real, positive numbers, and $a+b+c=1$. Prove the following:

$$ \frac{a^3}{a^2+b^2}+\frac{b^3}{b^2+c^2}+\frac{c^3}{c^2+a^2}\geq\frac{1}{2} $$

The only thing I could prove (using the Radon inequality) is that the left hand side is greater or equal with $\frac{1}{4}$.

Any ideas? I almost tried everything.