2024 AMC 12B 第 9 题

Question

一个飞镖靶是坐标平面中的区域

B

，由满足

|x| + |y| \le 8

的点

(x, y)

组成。目标区域

T

由满足

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

的点组成。飞镖随机落在

B

中一点。飞镖落在

T

中的概率可表示为

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

，其中

m

和

n

是互质正整数。求

m + n

。

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

答案：B

知识点：几何概率圆环

难度评级：1540

小提示：

区域

B

是对角线长为

16

的正方形，且

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

等价于

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

。

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

大提示：

环形区域

T

的面积为

\pi(32 - 18)

；检查其外半径

\sqrt{32}

恰好到达

B

的边，所以

T

完全位于

B

内。

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

解答：

飞镖靶

|x| + |y| \le 8

是一个对角线长为

16

的正方形，面积为

\tfrac12 \cdot 16 \cdot 16 = 128

。目标条件

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

等价于

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

，即

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

，这是一个面积为

\pi(32 - 18) = 14\pi

的圆环。

原点到正方形一条边（如

x + y = 8

）的距离为

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

，正好等于环形区域外半径，所以整个环形区域都在

B

内。概率为

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

，因此

m + n = 7 + 64 = 71

。

所以正确答案是 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.

2024 AMC 12B 第 9 题

答案：B

难度评级：1540

小提示：

大提示：

解答：

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