A1. The image shows three concentric semicircles with radii 1, 2, and 4. The areas marked with $X$ have an area of $x$, and the areas marked with $Y$ have an area of $y$. What is the ratio of the areas $x$ and $y$?
(A) $1: 3$
(B) $1: 2$
(C) $2: 3$
(D) $3: 8$
(E) $4: 9$

![](https://cdn.mathpix.com/cropped/2024_06_07_58dcb28c79ea4cc315e1g-10.jpg?height=248&width=454&top_left_y=864&top_left_x=1452)
A1. The areas of three semicircles are $\frac{\pi}{2}, \frac{4 \pi}{2}$, and $\frac{16 \pi}{2}$. The area of region $x$ is therefore $\frac{\frac{4 \pi}{2}-\frac{\pi}{2}}{2} = \frac{3 \pi}{4}$, and the area of region $y$ is $\frac{\frac{16 \pi}{2}-\frac{4 \pi}{2}}{3} = 2 \pi$. The desired ratio is thus $\frac{3 \pi}{4}: 2 \pi = 3: 8$.