AoPS - Problem

W.l.o.g. $y$ is the median of $\{x,y,z\}$. We have \begin{align*} 27(x^2y+y^2z+z^2x+xyz)\leq27(x^2y+y^2z+z^2x+xyz+z(x-y)(y-z))=4\cdot27y\left(\frac{x+z}{2}\right)^2\leq4(x+y+z)^3=4\\ 27(xy^2+yz^2+zx^2+xyz)\leq27(xy^2+yz^2+zx^2+xyz+x(x-y)(y-z))=4\cdot27y\left(\frac{x+z}{2}\right)^2\leq4(x+y+z)^3=4\\ 9(x^2y+y^2z+z^2x+2xy^2+2yz^2+2zx^2+3xyz)\leq4\\ (xy+yz+zx)(x+y+z)+xy^2+yz^2+zx^2\leq\frac{4}{9}\\ \frac{1}{xy+yz+zx+xy^2+yz^2+zx^2}\geq\frac{9}{4}\\ \frac x{y^2+z}+\frac y{z^2+x}+\frac z{x^2+y}=\frac{x^2}{xy^2+xz}+\frac{y^2}{yz^2+yx}+\frac{z^2}{zx^2+zy}\geq\frac{(x+y+z)^2}{xy+yz+zx+xy^2+yz^2+zx^2}\geq\frac{9}{4} \end{align*}