<s>[INST] Inequality with x,y,z fractions $\frac{x}{y}+\frac{y}{z+x}+\frac{z}{x}\ge 2$ If $x,y,z>0$, show:
$$\frac{x}{y}+\frac{y}{z+x}+\frac{z}{x}\ge 2$$
I expand and to prove
$$x^3 - 2 x^2 y + x^2 z + x y^2 - x y z + y z^2\ge 0$$
I don't know how to do this.
 [/INST] I found
$$x^3 - 2 x^2 y + x^2 z + x y^2 - x y z + y z^2 = xy(x-y)^2+x^2(z+x-y)^2+yz^2(y+z) \geqslant 0$$
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