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2024 AMC 10B Problem 14

Problem 14 of 25IntermediateGeometryCounting & Probability

A dartboard is the region BB in the coordinate plane consisting of points (x,y)(x, y) such that ∣x∣+∣y∣≤8.|x| + |y| \le 8. A target TT is the region where (x2+y2−25)2≤49.(x^2 + y^2 - 25)^2 \le 49. A dart is thrown and lands at a random point in B.B. The probability that the dart lands in TT can be expressed as mn⋅π,\dfrac{m}{n} \cdot \pi, where mm and nn are relatively prime positive integers. What is m+n?m + n?

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Solution

BB is the square ∣x∣+∣y∣≤8,|x| + |y| \le 8, with area 2⋅82=128.2 \cdot 8^2 = 128. The target condition (x2+y2−25)2≤49(x^2 + y^2 - 25)^2 \le 49 unpacks to ∣x2+y2−25∣≤7,|x^2 + y^2 - 25| \le 7, that is 18≤x2+y2≤32,18 \le x^2 + y^2 \le 32, an annulus of area π(32−18)=14π.\pi(32 - 18) = 14\pi. Does it fit inside B?B? The distance from the origin to an edge x+y=8x + y = 8 is 82=42=32,\tfrac{8}{\sqrt2} = 4\sqrt2 = \sqrt{32}, exactly the outer radius, so yes, the annulus sits inside the square. The probability is 14π128=764π,\tfrac{14\pi}{128} = \tfrac{7}{64}\pi, giving m+n=71.m + n = 71. Therefore, the answer is B.

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Concepts: geometric probability · annulus · tangent line

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