Problem:
Two circles of radius 1 are to be constructed as follows. The center of circle is chosen uniformly and at random from the line segment joining to . The center of circle is chosen uniformly and at random, and independently of the first choice, from the line segment joining to . What is the probability that circles and intersect?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Circles and both have radius 1 , so they intersect if and only if the distance between their centers is no greater than 2. Let the centers of the circles be and . The distance between these points is , so the circles intersect if and only if . This condition is equivalent to , or . Points in the square correspond to ordered pairs
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with and . The shaded region corresponds to the points that satisfy . Its area is . The requested probability is the area of the shaded region divided by the area of the square, which is
The problems on this page are the property of the MAA's American Mathematics Competitions