Using Cauchy-Schwarz we have: $ (y+ \sqrt{zx} + z )^2 \leq (x+y+z)(y+2z)$, so it remains to prove: $\sum \frac{2x^2+xy}{y+2z} \geq \sum x$ $\Leftrightarrow \sum \left( \frac{2x^2+xy}{y+2z}+x \right) \geq 2\sum x$ $\Leftrightarrow \sum \frac{x}{y+2z} \geq 1$ What follows from Cauchy-Schwarz: $\sum \frac{x}{y+2z} \geq \frac{(x+y+z)^2}{3(xy+xz+yz)} \geq 1$