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?

$0.3$

$0.4$

$0.5$

$0.6$

$0.7$

Answer: B

Concepts:geometric probability circle area lattice point

Difficulty rating: 1730

Small Hint:

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

Big Hint:

The four quarter-disks in one unit cell form one full disk of radius $d;$ set its area to $\tfrac12$

Solution:

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.

2020 AMC 12A Problem 16

Answer: B

Difficulty rating: 1730

Small Hint:

Big Hint:

Solution:

Problem 16 in Other Years