MathDB

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,

\overarc{AEB},\overarc{BFC}, and \overarc{CGD}, have their diameters on

\overline{AD},

and are tangent to line

EG

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]

<span class='latex-bold'>(A) </span> 13 \qquad<span class='latex-bold'>(B) </span> 14 \qquad<span class='latex-bold'>(C) </span> 15 \qquad<span class='latex-bold'>(D) </span> 16\qquad<span class='latex-bold'>(E) </span> 17

Problem Statement