Problem:
Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
Answer Choices:
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Solution:
Let be the center of the circle. Triangle is a right triangle, and is the midpoint of the hypotenuse . Then is a radius, and it is also one of the medians of the triangle. The centroid is located one third of the way along the median from to , so the centroid traces out a circle with center and radius (except for the two missing points corresponding to or ). The area of this smaller circle is then .
The problems on this page are the property of the MAA's American Mathematics Competitions