Task 2. Let's call a year interesting if a person turns as many years old as the sum of the digits of their birth year in that year. A certain year turned out to be interesting for Ivan, who was born in the 20th century, and for Vovochka, who was born in the 21st century. What is the difference in their ages?

Note. For convenience, we assume they were born on the same day, and all calculations are done in whole years.
Answer: 18 years.

Solution. Let Ivan's year of birth be $\overline{19 x y}$, and Vovochka's $-\overline{20 z t}$. An interesting year for Ivan will be $1900+10 x+y+10+x+y$, and for Vovochka $-2000+10 z+t+2+z+t$, according to the condition these values are equal, that is, $2002+11 z+2 t=1910+11 x+2 y$, from which $11(x-z)+2(y-t)=92$ or $11 a+2 b=92$, where the numbers $a=x-z$ and $b=y-t-$ are integers, ranging from -9 to 9.

We have $a=\frac{92-2 b}{11}$. The numerator here is an even number from 74 to 110, and it must be divisible by 11. Therefore, it is equal to 88 or 110. The second option does not fit, as $a \leq 9$. In the first case, we have $a=8 ; b=$ 2.

According to the condition of the problem, we need to find the number

$$
20 z t-19 x y=100-10(x-z)-(y-t)=100-10 a-b=18
$$

Let's check if this case is realized. For example, $8=9-1 ; 2=7-5$. Let Ivan be born in 1997, Vovochka - in 2015. For Ivan, the interesting year is $1997+1+9+9+7=2023$, for Vovochka $2015+2+0+1+5=2023$. At the same time, Ivan is older by $2015-1997=18$ years.

Note that we did not consider a special case when Ivan was born in 2000 (this year also belongs to the 20th century). In this case, the interesting year is 2002. But it will not be interesting for those born in 2001 (the interesting year is 2004), and even more so for subsequent years of birth (the interesting year comes after the year of birth).

Grading criteria: The correct answer (18) is given and one or several examples are provided when it is achieved - 5 points. An equation of the form $11 \mathrm{a}+2 \mathrm{~b}=92$ is obtained - 10 points. The complete solution of the previous equation ($a=8, b=2)$ - 5 points. A complete solution without considering the special case
(Ivan's year of birth is 2000) - 15 points. The special case is considered and it is shown that it does not fit - 5 points.