$\frac{x}{1 +\dfrac{yz}{x}}+ \frac{y}{1 + \dfrac{zx}{y}}+ \frac{z}{1 + \dfrac{xy}{z}} \geq \frac{(x+y+z)^2}{x+y+z+xy+yz+zx} \geq \frac{3(xy+yz+zx)}{xy+yz+zx+\sqrt{3(xy+yz+zx)}} =\frac{3(\sqrt{3}-1)}{2}$

parmenides51 wrote: Determine the largest possible value of$ M$ for which it holds that: $$\frac{x}{1 +\dfrac{yz}{x}}+ \frac{y}{1 + \dfrac{zx}{y}}+ \frac{z}{1 + \dfrac{xy}{z}} \ge M,$$for all real numbers $x, y, z > 0$ that satisfy the equation $xy + yz + zx = 1$. Greek TST 2013