2020 AMC 10A Problem 16

P16-20~4 minPrint

Geometric Probability Circles Bound the quantity above/below or estimate to pin it down Exploit symmetry to reduce work or pair up objects

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$ . The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$ . (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth?

Show solution without answering

Key insight

All unit squares look alike. The points within d of the four corners form four quarter-circles, one full circle of area pi d squared. Set that equal to 1/2.

Solution

The number $2020$ is only there to make the square big. Every $1 \times 1$ square in the grid looks exactly the same, so the probability is the same as for a single $1 \times 1$ square.

1. Take one unit square. Its lattice points are its four corners.
2. The points within $d$ of a corner form a quarter of a circle of radius $d$ inside the square.
3. All the answer choices are less than $0.5$ . With $d < 0.5$ , the quarter-circles from neighboring corners do not reach each other, so they do not overlap.
4. Four quarter-circles fit together into one whole circle. So the area within $d$ of a corner is $\pi d^2$ .
5. The unit square has area $1$ , so the probability is $\pi d^2$ . We want this to be $\frac12$ :
$\pi d^2 = \frac12, \qquad d^2 = \frac{1}{2\pi} \approx \frac{1}{6.28} \approx 0.159.$
6. Compare with the choices: $0.4^2 = 0.16$ , which is very close, while $0.3^2 = 0.09$ and $0.5^2 = 0.25$ are far off. So $d \approx 0.4$ .

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

Why this works

When a picture repeats in a grid, study one tile. Pieces of circles centered at the corners of a tile always add up to whole circles: four quarters make one. Just check that the pieces do not overlap before adding their areas.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(B)}\ 0.4}$ .

Review in the AoPS books

Introduction to Counting & Probability, Chapter 10: Geometric Probability
Introduction to Geometry, Chapter 11: Circles
Introduction to Algebra, Chapter 22: Special Manipulations (§22.3)
Introduction to Geometry, Chapter 12: Circles and Angles