Problem 1012

It is not hard to see, that Andriy's number can only be $\overline{100...0a}$, and Olesya's – only: $\overline{99...9b}$. Otherwise, the difference will not be a one-digit number. Really, if not all Olesya's digits, except for the last, are $9$, then after adding a one-digit number, the number of digits will not change. So this is the presentation of Andriy's number. The sum of digits of Olesya's number equals $2018$, so it is $\overline{99...92}$ (as $2018 = 224 \cdot 9 + 2$). So Andriy's number has to have the last digit less than $2$, because otherwise the difference of written numbers will not be less than $10$. So, this number can be $0$ or $1$. So, he wrote number $\overline{100...0}$, or $\overline{100...01}$.