USEMO 2019 Problem 3

Consider an infinite grid $\mathcal G$ of unit square cells. A chessboard polygon is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$ .

Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can Nikolai choose $F$ to make your job impossible?

Nikolai Beluhov