1. Let $a, b$ and $c$ be non-zero real numbers and

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

Calculate the value of the expression $a b+b c+c a$.
Solution. From the given equality, by direct calculation we obtain

$$
\begin{aligned}
& a+\frac{b}{c}=1 \Rightarrow a c+b=c \Rightarrow a c=c-b \\
& b+\frac{c}{a}=1 \Rightarrow a b+c=a \Rightarrow a b=a-c \\
& c+\frac{a}{b}=1 \Rightarrow c b+a=b \Rightarrow c b=b-a
\end{aligned}
$$

If we add the last equalities, we get

$$
a b+b c+c a=(c-b)+(a-c)+(b-a)=0
$$