Problem:
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?
Answer Choices:
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Solution:
Let the radius of the smaller circle be . Then the side length of the smaller square is . The radius of the larger circle is half the length of the diagonal of the smaller square, so it is . Hence the larger square has sides of length . The ratio of the area of the smaller circle to the area of the larger square is therefore
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions