58th Ukrainian National Mathematical Olympiad

It is not hard to see, that Andriy's number can only be $\overline{\mathrm{100...0}a}$, and Olesya's – only: $\overline{\mathrm{99...9}b}$. 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 $\overline{\mathrm{99...92}}$ (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 $\overline{\mathrm{100...0}}$, or $\overline{\mathrm{100...01}}$.