2024 AMC 10B Problem 14

Question

A dartboard is the region

B

in the coordinate plane consisting of points

(x, y)

such that

|x| + |y| \le 8.

A target

T

is the region where

(x^2 + y^2 - 25)^2 \le 49.

A dart is thrown and lands at a random point in

B.

The probability that the dart lands in

T

can be expressed as

\dfrac{m}{n} \cdot \pi,

where

m

and

n

are relatively prime positive integers. What is

m + n?

39

71

73

75

135

B

is a square of diagonal

16,

and

T

is the annulus

18 \le x^2 + y^2 \le 32

The outer radius

\sqrt{32} = 4\sqrt2

equals the distance from the origin to each edge of

B,

so

T

sits entirely inside

B

B

is the square

|x| + |y| \le 8,

with area

2 \cdot 8^2 = 128.

The target condition

(x^2 + y^2 - 25)^2 \le 49

unpacks to

|x^2 + y^2 - 25| \le 7,

that is

18 \le x^2 + y^2 \le 32,

an annulus of area

\pi(32 - 18) = 14\pi.

Does it fit inside

B?

The distance from the origin to an edge

x + y = 8

is

\tfrac{8}{\sqrt2} = 4\sqrt2 = \sqrt{32},

exactly the outer radius, so yes, the annulus sits inside the square. The probability is

\tfrac{14\pi}{128} = \tfrac{7}{64}\pi,

giving

m + n = 71.

Therefore, the answer is B.

Problem 14 in other years