1. Andrey and Kolya are not the same age, but last December each of them turned as many years old as the sum of the digits of their birth year. How old are they now?
Answer: 7 and 25.

## Solution.

Let the birth year be $\overline{a b c d}$, then $2021-\overline{a b c d}=a+b+c+d$, or $1001 a+101 b+$ $11 c+2 d=2021$. When $a=2$, we get $101 b+11 c+2 d=19$, that is, $b=0,11 c+2 d=$ 19. The digit $c$ must be odd and no more than 1, so $c=1$, then $d=4, \overline{a b c d}=$ 2014

When $a=1$, we get $101 b+11 c+2 d=1020,11 c+2 d \leq 99+18=117$, that is, $101 b \geq 903$, from which $b=9,11 c+2 d=111$. In this case, $11 c \geq 93$, so $d=9$. Then $d=6, \overline{a b c d}=1996$.

Verification. 1) $2021-2014=7$, the sum of the digits of the number 2014 is 7. 2) $2021-1996=25$, the sum of the digits of the number 1996 is 25.

Comment. Correct solution - 20 points. Points are distributed as follows: only one answer found - 7 points, the second answer found - 7 points, it is proven that there are no other solutions - 6 points, these points are added together. The solution is started, there is some progress - 5 points. The solution is started, but the progress is insignificant - 1 point. Only the answer without a solution - 1 point.