4. We will write down all five-digit numbers in which each of the digits $4,5,6,7,8$ appears exactly once. Then we will strike out one (any of them) and sum all the remaining ones. What are the possible values of the digital sum of such a result?

The regional round of category B takes place

## on Tuesday, April 5, 2011

so that it starts in the morning and the contestants have 4 hours of clean time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. Allowed aids are writing and drawing supplies, school MF tables, and calculators without a graphical display. These details will be communicated to the students before the start of the competition.
4. The final digit sum is uniquely determined and is the number 33.

To solve the problem, it will be advantageous to first determine the sum $S$ of all five-digit numbers containing each of the digits $4,5,6,7,8$. There are clearly as many such numbers as there are different permutations of the five digits, i.e., $5!=120$. Moreover, each of the given digits appears equally in each position among these 120 numbers, i.e., 24 times. The sum $S$ can thus be broken down by individual positions as

$$
\begin{aligned}
S= & 10^{4} \cdot(24 \cdot 4+24 \cdot 5+24 \cdot 6+24 \cdot 7+24 \cdot 8)+ \\
& +10^{3} \cdot(24 \cdot 4+24 \cdot 5+24 \cdot 6+24 \cdot 7+24 \cdot 8)+\ldots= \\
= & 24 \cdot(4+5+6+7+8) \cdot\left(10^{4}+10^{3}+10^{2}+10+1\right)=24 \cdot 30 \cdot 11111 .
\end{aligned}
$$

Let us now turn our attention to the possible values of the digit sum of the number $S-a$, where $a$ is a five-digit number of the mentioned form, i.e., $a=33333+b$, where $b$ is a five-digit number containing each of the digits $1,2,3,4,5$. Thus,

$$
S-a=11111 \cdot 24 \cdot 30-a=7999920-33333-b=7966587-b .
$$

However, when subtracting the number $b$, there is no carry-over in the individual positions, so the digit sum of the number $S-a$ is equal to $(7+9+6+6+5+8+7)-(1+2+3+4+5)=48-15=33$ for any five-digit number $a$ containing each of the digits $4,5,6,7,8$.

For a complete solution, award 6 points.