2024 AMC 12B Problem 9

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

The region

B

is a square with diagonals of length

16,

and

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

means

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

The annulus

T

has area

\pi(32 - 18);

check that its outer radius

\sqrt{32}

exactly reaches the sides of

B,

so

T

lies entirely inside

B

The dartboard

|x| + |y| \le 8

is a square with diagonals

16,

so its area is

\tfrac12 \cdot 16 \cdot 16 = 128.

The target condition

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

means

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

i.e.

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

an annulus of area

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

The distance from the origin to a side of the square (for instance

x + y = 8

) is

\dfrac{8}{\sqrt2} = \sqrt{32},

exactly the annulus’s outer radius. So the annulus is tangent to the square and lies entirely within

B.

The probability is

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

giving

m + n = 7 + 64 = 71.

Thus, the correct answer is B.

Problem 9 in other years