Szilveszter noticed that if he adds the digits of his birth year, he gets the number formed by the last two digits of his grandfather's birth year. By combining the last two digits of Szilveszter's birth year, his grandfather's current age is obtained. How old is Szilveszter?
Szilveszter's grandfather cannot be older than 99, as his current age can be written with a two-digit number. Thus, both he and Szilveszter were definitely born in the 1900s. (The problem was posed in 1999.) Let Szilveszter's birth year be $\overline{19 a b}$. Then his grandfather was born in the year $1900+1+9+a+b$, so he is $1999-1900-1-9-a-b=89-a-b$ years old. We know that this is equal to $\overline{a b}$, so

$$
\overline{a b}=89-a-b, 10 a+b=89-a-b, 11 a+2 b=89
$$

where $a$ and $b$ are both single-digit numbers.

$2 b$ can be at most 18, so $11 a$ must be at least 71, meaning $a$ can only be 7 or 8. (If $a=9$, then $11 a$ is greater than 89).

For $a=7$, then $2 b=12$, from which $b=6$, so Szilveszter was born in 1976, his grandfather in 1900+1+9+7+6=1923, so he is now 76 years old, which is indeed the last two digits of Szilveszter's birth year.

If $a=8$, then $2 b=1, b$ is not an integer, so we do not get a solution.

Szilveszter was born in 1976, and is now 23 years old.