a) Let us show that Vasyl' can achieve the situation that after each his move, the board contains some odd number $n$ and $2019-n$ numbers $1$. Then, after $1009$ pairs of moves, only number $2019$ remains on the board, and Petryk will not be able to make a move and will lose. As his first move, Petryk wipes two numbers $1$ and writes down number $2$. Vasyl' makes number $3$. If at some point an odd number $n$ is written on the board, and $2019-n$ numbers $1$, then Petryk can either erase two $1$ and write $2$, after which Vasyl' erases $n$ and $2$ and writes $n+2$, or he exchanges $n$ and $1$ for $n+1$, after which Vasyl' erases $n+1$ and $1$ and writes $n+2$, and wins.

b) Here Vasyl' follows a strategy similar to a) except for the last move. Before the last move of Petryk, four numbers are written on the board: $2017$, $1$, $1$ and $1$. If Petryk leaves the numbers $2018$, $1$ and $1$, then Vasyl' turns it into $2018$ and $2$, and Petryk cannot make a move. If Petryk leaves the numbers $2017$, $2$ and $1$, then Vasyl' again turns it into $2018$ and $2$, and Petryk cannot make a move.