1. Find a three-digit number with distinct digits, which is five times smaller than the sum of the other three-digit numbers written with the same digits. Determine all solutions!
Solution. One

$$
5 \overline{x y z}=\overline{x z y}+\overline{y x z}+\overline{y z x}+\overline{z x y}+\overline{z y x}
$$

i.e.

$$
6 \overline{x y z}=\overline{x y z}+\overline{x z y}+\overline{y x z}+\overline{y z x}+\overline{z x y}+\overline{z y x}
$$

we obtain

$$
\begin{aligned}
& 6(100 x+10 y+z)=222(x+y+z) \\
& 7 x=3 y+4 z \\
& 7(x-z)=3(y-x)
\end{aligned}
$$

According to this, $7 \mid(y-z)$ and since $z$ and $y$ are single-digit numbers, we get

$$
(y, z) \in\{(8,1),(1,8),(9,2),(2,9)\}
$$

By substituting in the last equation, we obtain the numbers 481, 518, 592, and 629.