MathDB

Circles

\mathcal{C}_1

\mathcal{C}_2

, and

\mathcal{C}_3

have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line

t_1

is a common internal tangent to

\mathcal{C}_1

and

\mathcal{C}_2

and has a positive slope, and line

t_2

is a common internal tangent to

\mathcal{C}_2

and

\mathcal{C}_3

and has a negative slope. Given that lines

t_1

and

t_2

intersect at

(x,y)

, and that

x=p-q\sqrt{r}

, where

p

q

, and

r

are positive integers and

r

is not divisible by the square of any prime, find

p+q+r

Problem Statement