Problem:
Two concentric circles have radii and . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
Answer Choices:
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B.
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E.
Solution:
Let be the first point chosen on the outer circle, let chords and on the outer circle be tangent to the inner circle at and , respectively, and let be the common center of the two circles. Triangle has a right angle at , and , so . Similarly, , so , and minor . If is the second point chosen on the outer circle, then chord intersects the inner circle if and only if is on minor . Therefore the requested probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions