\(x+y+z=1\\ \Rightarrow\left\{{}\begin{matrix}x=1-y-z\\y=1-x-z\\z=1-x-y\end{matrix}\right.\)
\(S=\dfrac{\left(xy+z\right)\left(yz+x\right)\left(zx+y\right)}{\left(1-x\right)^2\left(1-y\right)^2\left(1-z\right)^2}\)
\(\Rightarrow S=\dfrac{\left(xy+1-x-y\right)\left(yz+1-y-z\right)\left(zx+1-x-z\right)}{\left(x+y+z-x\right)^2\left(x+y+z-y\right)^2\left(x+y+z-z\right)^2}\)
\(\Rightarrow S=\dfrac{\left[\left(xy-x\right)-\left(y-1\right)\right]\left[\left(yz-y\right)-\left(z-1\right)\right]\left[\left(zx-x\right)-\left(z-1\right)\right]}{\left(y+z\right)^2\left(x+z\right)^2\left(x+y\right)^2}\)
\(\Rightarrow S=\dfrac{\left[x\left(y-1\right)-\left(y-1\right)\right]\left[y\left(z-1\right)-\left(z-1\right)\right]\left[x\left(z-1\right)-\left(z-1\right)\right]}{\left(y+z\right)^2\left(x+z\right)^2\left(x+y\right)^2}\)
\(\Rightarrow S=\dfrac{\left(x-1\right)\left(y-1\right)\left(y-1\right)\left(z-1\right)\left(x-1\right)\left(z-1\right)}{\left(y+z\right)^2\left(x+z\right)^2\left(x+y\right)^2}\)
\(\Rightarrow S=\dfrac{\left(x-1\right)^2\left(y-1\right)^2\left(z-1\right)^2}{\left(y+z\right)^2\left(x+z\right)^2\left(x+y\right)^2}\)
\(\Rightarrow S=\dfrac{\left(x-x-y-z\right)^2\left(y-x-y-z\right)^2\left(z-x-y-z\right)^2}{\left(y+z\right)^2\left(x+z\right)^2\left(x+y\right)^2}\)
\(\Rightarrow S=\dfrac{\left(-y-z\right)^2\left(-x-z\right)^2\left(-x-y\right)^2}{\left(y+z\right)^2\left(x+z\right)^2\left(x+y\right)^2}\)
\(\Rightarrow S=\dfrac{\left(y+z\right)^2\left(x+z\right)^2\left(x+y\right)^2}{\left(y+z\right)^2\left(x+z\right)^2\left(x+y\right)^2}\)
\(\Rightarrow S=1\)
