Problem:
In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let be the combined area of the small semicircles and be the area of the region inside the large semicircle but outside the small semicircles. The ratio is . What is ?
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Solution:
Suppose without loss of generality that each small semicircle has radius 1 ; then the large semicircle has radius . The area of each small semicircle is , and the area of the large semicircle is . The combined area of the small semicircles is , and the area inside the large semicircle but outside the small semicircles is
Thus the ratio of the areas is , which is . Because this ratio is given to be , it follows that and .
The problems on this page are the property of the MAA's American Mathematics Competitions