2020 AMC 12A Problem 16

Question

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

\tfrac12.

(A point

(x, y)

is a lattice point if

x

and

y

are both integers.) What is

d

to the nearest tenth?

0.3

0.4

0.5

0.6

0.7

By periodicity, focus on a single unit square with a lattice point at each corner

The four quarter-disks in one unit cell form one full disk of radius

d;

set its area to

\tfrac12

By periodicity it suffices to consider one unit cell with a lattice point at each corner. The region within

d

of a corner consists of four quarter-disks of radius

d,

forming one full disk of area

\pi d^2.

The resulting value will be less than

\tfrac12,

so these quarter-disks do not overlap.

Setting

\pi d^2 = \tfrac12

gives

d = \sqrt{\dfrac{1}{2\pi}} \approx 0.399.

To the nearest tenth,

d = 0.4.

Thus, B is the correct answer.

Problem 16 in other years