1. Determine the value of the expression

$$
S=\frac{1}{1+x+x y}+\frac{1}{1+y+y z}+\frac{1}{1+z+z x}
$$

if for the real numbers $x, y, z$ it holds that $x y z=1$.
Solution. We have

$$
\begin{aligned}
S & =\frac{1}{1+x+x y}+\frac{1}{1+y+y z}+\frac{1}{1+z+z x} \\
& =\frac{z}{z+x z+x y z}+\frac{x z}{x z+x y z+x y z^{2}}+\frac{1}{1+z+z x} \\
& =\frac{z}{1+z+x z}+\frac{x z}{1+z+x z}+\frac{1}{1+z+z x}=1
\end{aligned}
$$

provided that the expression $S$ is defined.