We know that the product $a b c$ is equal to 1. What is the value of the following expression:

$$
\frac{a}{a b+a+1}+\frac{b}{b c+b+1}+\frac{c}{c a+c+1}
$$
We can extend the second fraction by $a$, and the third one by $a b$. Then we get the sum in the following form:

$$
\frac{a}{a b+a+1}+\frac{a b}{a b c+a b+a}+\frac{a b c}{a b c \cdot a+a b c+a b}
$$

Using the fact that $a b c=1$, the denominators of the fractions become equal, and the sum of the numerators is exactly equal to the denominators. Therefore, the value of the expression is 1, provided that the fractions are defined.