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2024 AMC 12B Problem 9

Problem 9 of 25EasierGeometryCounting & 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

The dartboard ∣x∣+∣y∣≤8|x| + |y| \le 8 is a square with diagonals 16,16, so its area is 12⋅16⋅16=128.\tfrac12 \cdot 16 \cdot 16 = 128. The target condition (x2+y2−25)2≤49(x^2 + y^2 - 25)^2 \le 49 means −7≤x2+y2−25≤7,-7 \le x^2 + y^2 - 25 \le 7, i.e. 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. The distance from the origin to a side of the square (for instance x+y=8x + y = 8) is 82=32,\dfrac{8}{\sqrt2} = \sqrt{32}, exactly the annulus’s outer radius. So the annulus is tangent to the square and lies entirely within B.B. The probability is 14π128=764π,\dfrac{14\pi}{128} = \dfrac{7}{64}\pi, giving m+n=7+64=71.m + n = 7 + 64 = 71. Thus, the correct answer is B.

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Concepts: geometric probability · annulus

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.