A1. In a circle with diameter 4, we inscribe a square, in the resulting square we inscribe a circle, and in this one again a square, and we repeat the process. What is the diameter of the fourth circle?
(A) $\frac{1}{\sqrt{2}}$
(B) $\frac{1}{2}$
(C) 1
(D) $\sqrt{2}$
(E) $2 \sqrt{2}$
A1. The ratio of the diameters of two consecutive circles is equal to the ratio between the diagonal and the side of a square, that is $\sqrt{2}$. Therefore, the ratio of the diameters between the first and fourth circle is equal to $(\sqrt{2})^{3}=2 \sqrt{2}$. The diameter of the fourth circle is thus equal to $\frac{4}{2 \sqrt{2}}=\sqrt{2}$.