Problem:
A dartboard is the region in the coordinate plane consisting of points such that . A target is the region where . A dart is thrown and lands at a random point in . The probability that the dart lands in can be expressed as , where and are relatively prime positive integers. What is ?
Answer Choices:
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Solution:
The region is a square with intercepts and . The area of this square is . Taking square roots shows that region is the set of points that satisfy
which is an annulus (ring) with inner radius and outer radius . Its area is . Note that is internally tangent to at the four points . The required probability is the ratio of the area of to the area of , namely . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions