Line segment $\overline{AB}$ s a diameter of a circle with $AB = 24.$ Point $C,$ not equal to $A$ or $B,$ lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?

38
63
50
75
25

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The point has to be on a line connecting C to the middle of AB, that is to the center O of the circle.

It also has to be located $\frac13$ of the distance OC from O.

The curve is therefore a circle centered at O with a radius $\frac13$ of the radius of the original circle, that is $r=\frac12\cdot\frac13\cdot24=4$ .

The area of this circle is then $A=16\pi\approx\boxed{50}$ .