oVlad wrote: Let $x,y,z$ be positive real numbers. Prove that \[\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\geqslant\frac{z(x+y)}{y(y+z)}+\frac{x(y+z)}{z(z+x)}+\frac{y(z+x)}{x(x+y)}.\] Moldova TST 2013 $$\iff \left( \frac{y}{z}-\frac{x(z+y)}{z(z+x)} \right) + \left(\frac{z}{x}\ -\frac{y(x+z)}{x(x+y)} \right) + \left(\frac{x}{y} -\frac{z(y+x)}{y(y+z)} \right) \geq 0\iff \frac{y-x}{z+x} + \frac{z-y}{z+y} +\frac{z-y}{x+y} \geq0$$$$\iff \left(\frac{y-x}{z+x} +1\right) + \left(\frac{z-y}{x+y} +1\right)+ \left( \frac{x-z}{y+z} +1\right) \geq 3 \iff \frac{y+z}{z+x} +\frac{z+x}{x+y} +\frac{z+x}{x+y} \geq 3$$https://artofproblemsolving.com/community/c4h2429811p20053043