> 
> Let $a$, $b$ and $c$ be positive numbers. Prove that:
>  $$\frac{a^3}{13a^2+5b^2}+\frac{b^3}{13b^2+5c^2}+\frac{c^3}{13c^2+5a^2}\geq\frac{a+b+c}{18}$$
> 
> 
> 

This inequality is strengthening of the following Vasile Cirtoaje's one, which he created in 2005.

> 
> Let $a$, $b$ and $c$ be positive numbers. Prove that:
>  $$\frac{a^3}{2a^2+b^2}+\frac{b^3}{2b^2+c^2}+\frac{c^3}{2c^2+a^2}\geq\frac{a+b+c}{3}.$$
> 
> 
> 

My proof of this inequality you can see here: <https://artofproblemsolving.com/community/c6h22937p427220>

But this way does not help for the starting inequality.

A big problem we have around the point $(a,b,c)=(0.785, 1.25, 1.861)$ because the difference between the LHS and the RHS in this point is $0.0000158...$.

I tried also to use Cauchy-Schwarz, but without success.

Also, I think the BW (see here <https://math.stackexchange.com/tags/buffalo-way/info> I tryed!) does not help.