## Task B-1.5.

On the way to the sports hall, Marin asked Ivan for the number of his seat. Wanting to reduce the tension before the important sports event and entertain his friend, Ivan enigmatically replied: "All the digits of that number are different. If you add up all six two-digit numbers that can be formed from the digits of that number, so that each of those two-digit numbers also has different digits, and then divide the obtained sum by two, you will get the number of my seat." What is Ivan's seat number?
## Solution.

Six two-digit numbers with different digits can only be formed from three different digits.

Therefore, the required number must be a three-digit number $N$ with different digits, none of which is 0.

Let the digits of this number be $x, y$, and $z$. Then,

$$
N=100 x+10 y+z
$$

and the corresponding two-digit numbers are

$$
10 x+y, 10 x+z, 10 y+z, 10 y+x, 10 z+x \text { and } 10 z+y
$$

From the condition of the problem, it follows that

$$
\begin{aligned}
& \frac{1}{2}(10 x+y+\ldots+10 z+y)=100 x+10 y+z \\
& 11(x+y+z)=100 x+10 y+z \\
& y+10 z=89 x
\end{aligned}
$$

On the left side, we have a two-digit number, which means that the digit $x$ can only be 1, i.e., $x=1$.

1 point

From the equation $y+10 z=89$, it follows that $z=8$ and $y=9$.

1 point

The required number of seats is 198.