Let $a, b, c$ be positive real numbers such that $a b c=1$. Simplify

$$
\frac{1}{1+a+a b}+\frac{1}{1+b+b c}+\frac{1}{1+c+c a} .
$$
We may let $a=y / x, b=z / y, c=x / z$ for some real numbers $x, y, z$. Then

$$
\begin{aligned}
\frac{1}{1+a+a b}+\frac{1}{1+b+b c}+\frac{1}{1+c+c a} & =\frac{1}{1+y / x+z / x}+\frac{1}{1+z / y+x / y}+\frac{1}{1+x / z+y / z} \\
& =\frac{x}{x+y+z}+\frac{y}{x+y+z}+\frac{z}{x+y+z} \\
& =1 .
\end{aligned}
$$