2020 AMC 10A Problem 16

The big square is tiled by $2020^2$ unit squares whose corners are the lattice points, and each unit square looks exactly the same. So the probability equals the fraction of a single unit square lying within $d$ of one of its four corners.

Since the answer choices are all below $0.5 < 1$, the quarter-discs of radius $d$ at the four corners do not overlap (they would need $d \ge \frac12$ to touch). Four quarter-discs of radius $d$ have the same total area as one full disc, $\pi d^2$.

Setting the covered fraction equal to $\frac12$:
$$\pi d^2 = \frac12 \quad\Longrightarrow\quad d^2 = \frac{1}{2\pi} \approx \frac{1}{6.28} \approx 0.159,$$
so $d \approx 0.40$ (since $0.4^2 = 0.16$).

The answer is $\boxed{\textbf{(B)}\ 0.4}$.

Periodic pictures reduce to one fundamental tile: the $2020$ is a distraction. The four corner pieces of a disc re-form a whole disc, a standard trick for anything centered at lattice points. Always check the non-overlap condition before trusting the reassembly; here $d < 0.5$ guarantees it.

• Counting a full disc at each of the four corners of a unit square (4 pi d^2 = 1/2) or only one quarter-disc, which changes d by a factor of 2.
• Stopping at $d^2 \approx 0.16$ and forgetting the square root, or letting the size $2020$ suggest boundary effects matter.