59th Ukrainian National Mathematical Olympiad

Firstly, we will show that the number on the board decreases with every turn. Clearly, it will be smaller and integer. It will be positive, since: $N=dD\Rightarrow M=N-(2d-1)=dD-2d+1=d(D-2)+1\ge 1$, since if $d<N$ then $D\ge 2$. Therefore, number $1$ will be written on the board after finite number of turns.

It isn't hard to notice, that every turn changes the parity of the number on the board. Since Katia starts the game, there will be even number after her turn, and odd number after Mykola's turn. Therefore, number $1$ will be written on the board after Mykola's turn, so he will lose.