HMMT-Nov 2019 thm, Problem 2

Sandy likes to eat waffles for breakfast. To make them, she centers a circle of waffle batter of radius $3 \mathrm{~cm}$ at the origin of the coordinate plane and her waffle iron imprints non-overlapping unit-square holes centered at each lattice point. How many of these holes are contained entirely within the area of the waffle?

Solution: First, note that each divet must have its sides parallel to the coordinate axes; if the divet centered at the lattice point $(a, b)$ does not have this orientation, then it contains the point $(a+1 / 2, b)$ in its interior, so it necessarily overlaps with the divet centered at $(a+1, b)$. If we restrict our attention to one quadrant, we see geometrically that the divets centered at $(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0)$, and $(2,1)$ are completely contained in the waffle, and no others are. We can make this more rigorous by considering the set of points $(x, y)$ such that $x^{2}+y^{2}<9$. We count 1 divet centered at the origin, 8 divets centered on the axes that are not centered at the origin, and 12 divets not centered on the axes, for a total of 21 divets. $\fbox{21}$.

Answer: 21