Mind The Gaps

With Descartes' Circle Theorem in hand, this problem is relatively straightforward. The theorem states that if four circles are mutually tangent (each one tangent to the other three), their curvatures $k_1, k_2, k_3, k_4$ satisfy the equation

$(k_1+k_2+k_3+k_4)^2=2(k_1^2+k_2^2+k_3^2+k_4^2).$

(The curvature of a circle is the reciprocal of its radius, and if three of the four circles are inside the fourth, the outer circle is said to have negative curvature.)

If we decide that the largest circle has a radius of 1, then we'll consider its curvature to be $-1$ . Then the two largest gray circles would each have a curvature of 2. Plugging $-1$ , $2$ , and $2$ into Descartes' equation and solving for the fourth curvature yields a curvature of 3 for the gray circle above the red one.

Plug in $2$ , $2$ , and $3$ and solve, and you get a curvature of 15 for the red circle.

Plug in $2$ , $3$ , and $-1$ and solve, and you get a curvature of 6 for the smallest gray circle; do the same with $-1$ , $6$ , and $3$ to yield a curvature of 14 for the blue circle. So the desired ratio is

$\frac{\text{blue radius}}{\text{red radius}}=\frac{\frac{1}{14}}{\frac{1}{15}}=\frac{15}{14},$

and the answer is $15+14=\boxed{29}$ .