3 fractions

We have $\frac1{1+x+xy}+\frac1{1+y+yz}+\frac1{1+z+zx}$ In the first fraction, multiply numerator and denominator by $z$ , we get $\frac{z}{z+xz+1}+\frac1{1+y+yz}+\frac1{1+z+zx}=\frac1{1+y+yz}+\frac{1+z}{1+z+zx}$ Now in the second term, multiply $N^m, D^m$ by $y$ , we get $\frac{1}{1+y+yz}+\frac{y+yz}{y+yz+1}=\frac{1+yz+y}{1+yz+y}=1$

Put $x=y=z=1$ , $\dfrac{1}{1+x+xy}+\dfrac{1}{1+y+yz} +\dfrac{1}{1+z+zx}=3\cdot\dfrac{1}{3}=1$ .

$xyz = 1$

Count it again and you will find out that $x = y = z = 1$

So $\frac {1}{3} \times 3 = 1$