International Mathematical Olympiad (IMO) $2018$ , Day $2$ , Problem $4$ of $6$

A site is any point $x, y$ in the plane such that $x$ and $y$ are both positive integers $\leq 20$ . Initially, each of the $400$ sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\surd 5$ . On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.

The person that answers correctly and gives the official solution first - go to International Mathematical Olympiad (IMO) Hall of Fame