1. Find all triples of real numbers $(u ; v ; w)$, satisfying the system of equations:

$$
\left\{\begin{array}{l}
u+v \cdot w=20 \\
v+w \cdot u=20 \\
w+u \cdot v=20
\end{array}\right.
$$
Answer: $(4 ; 4 ; 4),(-5 ;-5 ;-5),(1 ; 1 ; 19),(1 ; 19 ; 1),(19 ; 1 ; 1)$.

Solution: Subtract the first equation from the second

$$
\begin{gathered}
(v-u)-w(v-u)=0 \\
(v-u)(1-w)=0
\end{gathered}
$$

1) $v=u \Rightarrow\left\{\begin{array}{c}v+w \cdot v=20, \\ w+v^{2}=20 .\end{array} \Rightarrow(w-v)-v(w-v)=0 \Rightarrow(w-v)(1-v)=0\right.$

From which it follows that $v=u=w$ or $v=u=1$.

If $v=u=w$, then $\quad v=u=w=4$ or $v=u=w=-5$.

If $v=u=1$, then $\quad w=19$.
2) $w=1 \Rightarrow\left\{\begin{array}{l}v+u=20 \\ u \cdot v=19 .\end{array} \Rightarrow\left[\begin{array}{l}v=19, u=1 \\ v=1, u=19\end{array}\right.\right.$