AMC 10 B

Geometry Difficulty 4.4 AIME Find the answer United States

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 $\frac{m}{n} \cdot \pi$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?

Pick one

Solution

Answer (B): The region $B$ is a square with intercepts $(\pm8, 0)$ and $(0, \pm8)$ . The area of this square is $(8\sqrt{2})^2 = 128$ . Taking square roots shows that region $T$ is the set of points that satisfy
$25 - 7 \le x^2 + y^2 \le 25 + 7,$
which is an annulus (ring) with inner radius $\sqrt{18}$ and outer radius $\sqrt{32}$ . Its area is $32\pi - 18\pi = 14\pi$ . Note that $T$ is internally tangent to $B$ at the four points $(\pm4, \pm4)$ . The required probability is the ratio of the area of $T$ to the area of $B$ , namely $\frac{14\pi}{128} = \frac{7}{64}\pi$ . The requested sum is $7 + 64 = 71$ .

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