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Slovenia — National Math Olympiad, 2012

Geometry Difficulty 3.4 AMC 10/12 Find the answer Slovenia

In the first circle with diameter $4$ we inscribe a square. In the square we inscribe a new circle. In the last circle we again inscribe a square and repeat the process. What is the diameter of the fourth circle?

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Solution

The ratio of the diameters of two consecutive circles is equal to the ratio of the diagonal and the side of the square, hence $\sqrt{2}$ . The ratio of the diameters of the first and the fourth circle is thus equal to $(\sqrt{2})^3 = 2\sqrt{2}$ . The diameter of the fourth circle is $\frac{4}{2\sqrt{2}} = \sqrt{2}$ .

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