IMC 2013 Q22

Let the radius of each semicircle be \(r\). In the left-hand diagram, let the side-length of the square be 2\(x\). By Pythagoras' Theorem, \(x^2 + (2x)^2 = r^2\) and so \(5x^2 = r^2\). So this shaded area is \(4x^2 = \tfrac{4r^2}{5}\). In the right-hand diagram, let the side-length of each square be \(y\). Then by Pythagoras' Theorem, \(y^2 + y^2 = r^2\) and so this shaded area is \(r^2\).
Therefore the ratio of the two shaded areas is \(\tfrac{4}{5}\) : 1 = 4 : 5.