9.4. To the number $A$, consisting of eight non-zero digits, a seven-digit number, consisting of identical digits, was added, and the eight-digit number $B$ was obtained. It turned out that the number $B$ can be obtained from the number $A$ by rearranging some of the digits. What digit can the number $A$ start with if the last digit of the number $B$ is 5?
Answer: 5.

Solution: Since the numbers $A$ and $B$ have the same sum of digits, their difference is divisible by 9. Therefore, the added seven-digit number with identical digits is divisible by 9. This means it consists of nines. That is, we can consider that $10^7$ was added to the number $A$ and 1 was subtracted. This means that the number $B$ is obtained from the number $A$ by increasing the first digit by 1 and decreasing the last digit by 1 (since there are no zeros in $A$ and $B$ is eight-digit), while the other digits remain unchanged. Since the number $B$ can be obtained from the number $A$ by permuting some digits, the last digit of the number $B$ coincides with the first digit of the number $A$ (and vice versa). Therefore, the number $A$ can only start with the digit 5.

Comment: It is proven that the added number consists of nines - 2 points.

Answer obtained by considering an example - 1 point.