USAJMO 2023 Problem 4

Two players, $B$ and $R$ , play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with $B$ . On $B$ 's turn, $B$ selects one white unit square and colors it blue. On $R$ 's turn, $R$ selects two white unit squares and colors them red. The players alternate until $B$ decides to end the game. At this point, $B$ gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score $B$ can guarantee?

(A simple polygon is a polygon (not necessarily convex) that does not intersect itself and has no holes.)

Proposed by David Torres