58th Ukrainian National Mathematical Olympiad

Notice that $n$ cannot be represented as a difference of squares of integers iff $n\equiv 2(\mathrm{mod}4)$. Indeed, if $n\equiv 2(\mathrm{mod}4)$, then suppose $n=(x+y)(x-y)$. If $x,y$ are either both odd or both even, then $n\equiv 0(\mathrm{mod}4)$, otherwise $n\equiv \pm 1(\mathrm{mod}4)$. If $n=4k$, then let $x=k+1,y=k-1$, if $n\equiv \pm 1(\mathrm{mod}4)$, then let $x=\frac{n+1}{2},y=\frac{n-1}{2}$.

The strategy for Petryk is the following: he erases $100$ first, then if Ivasyk chooses $m$, then Petryk erases $100-m$. Since choice of number $50$ leads to loss, then Petryk wins, because if Ivasyk hasn't lost yet then Petryk has a number to erase that will not make him lose.