2025 AMC 10B Problem 20

Four congruent semicircles are inscribed in a square of side length 1 so that their diameters are on the sides of the square, one endpoint of each diameter is at a vertex of the square, and adjacent semicircles are tangent to each other. A small circle centered at the center of the square is tangent to each of the four semicircles, as shown below.

The diameter of the small circle can be written as (\sqrt{a} + b)(\sqrt{c} + d), where a, b, c, and d are integers. What is a + b + c + d?

\textbf{(A)} 3 \qquad \textbf{(B)} 5 \qquad \textbf{(C)} 8 \qquad \textbf{(D)} 9 \qquad \textbf{(E)} 11

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.