Problem
2019 AMC 12B Problem 15
As shown in the figure, line segment \overline{AD} is trisected by points B and C so that AB=BC=CD=2. Three semicircles of radius 1, \text{arc }AEB,\text{arc }BFC, and \text{arc }CGD, have their diameters on \overline{AD}, and are tangent to line EG at E,F, and G, respectively. A circle of radius 2 has its center on F. The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form \frac{a}{b}\cdot\pi-\sqrt{c}+d,where a,b,c, and d are positive integers and a and b are relatively prime. What is a+b+c+d?
![[asy] size(6cm); filldraw(circle((0,0),2), gray(0.7)); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label("$A$",(-3,-1),S); dot((-2,0)); label("$E$",(-2,0),NW); dot((-1,-1)); label("$B$",(-1,-1),S); dot((0,0)); label("$F$",(0,0),N); dot((1,-1)); label("$C$",(1,-1), S); dot((2,0)); label("$G$", (2,0),NE); dot((3,-1)); label("$D$", (3,-1), S); [/asy]](/static/aops/be5538c17cf2af3d37027f4996d5a8ae5cd4aef1.png)
\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17
Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.
Want to contribute problems and receive full credit? Click here to add your problem!
Please report any issues to us in our Discord server
Go to previous contest problem (SHIFT + Left Arrow) Go to next contest problem (SHIFT + Right Arrow)
Problem feedback
Difficulty
—