$x,y,z > 0$, prove $$\sum_{\text{cyc}}\frac{x(y-z)}{(2x+y)^2} +\frac13 \cdot \frac{x^2+y^2+z^2}{xy+yz+zx} \geqslant \frac13$$
While this inequality can be proved by brute force, the elegant solution has never been found. I think we can rewrite the inequality as a sum of squares, but the cyclic nature made it very difficult.
Please help.
