On a clean board, we wrote a three-digit natural number with yellow chalk, composed of mutually different non-zero digits. Then we wrote on the board with white chalk all other three-digit numbers that can be obtained by changing the order of the digits of the yellow number. The arithmetic mean of all the numbers on the board was 370. We underlined each number smaller than the yellow one. The underlined numbers were exactly three and their arithmetic mean was 205. Determine the yellow number.

(L. Šimünek)
The digits of the sought number will be denoted by $a, b, c$, assuming

$$
0<a<b<c \text {. }
$$

The numbers on the board, arranged from the smallest to the largest, are

$$
\overline{a b c}, \overline{a c b}, \overline{b a c}, \overline{b c a}, \overline{c a b}, \overline{c b a}
$$

The arithmetic mean of the first three of them is 205, so we have

$$
(100 a+10 b+c)+(100 a+10 c+b)+(100 b+10 a+c)=3 \cdot 205
$$

after simplification

$$
210 a+111 b+12 c=615 .
$$

Similarly, we derive the equation based on the knowledge of the arithmetic mean of all the numbers:

$$
222 a+222 b+222 c=6 \cdot 370,
$$

after simplification

$$
\begin{aligned}
222 \cdot(a+b+c) & =2220 \\
a+b+c & =10
\end{aligned}
$$

From the derived relationships (1), (2), and (3), we are now able to determine the digits $a, b$, and $c$ uniquely: For $a \geqq 3$, the value on the left side of equation (2) would be too large, and this equality could not hold. For $a=2$, given condition (1), $b \geqq 3$ would be required, and even in such a case, the value on the left side of equation (2) would be too large. Therefore, $a=1$. Substituting into equations (2) and (3) and simplifying, we obtain the system

$$
\begin{aligned}
111 b+12 c & =405 \\
b+c & =9
\end{aligned}
$$

which has the unique solution $b=3, c=6$. The yellow number, written as $\overline{b c a}$, is therefore 361.

Evaluation. 2 points for determining that the digit sum is $10$; 2 points for the relationship $210 a+111 b+12 c=615$ or its equivalent; 1 point for determining the digits of the sought number; 1 point for the sought number.