HMMT 十一月 2019 · THM 赛 · 第 2 题

HMMT November 2019 — THM Round — Problem 2

Sandy likes to eat waffles for breakfast. To make them, she centers a circle of waffle batter of radius 3cm 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?

解析

Sandy likes to eat waffles for breakfast. To make them, she centers a circle of waffle batter of radius 3cm 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? Proposed by: Carl Schildkraut Answer: 21 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 2 2 x + y < 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.