Christmas Contest - Team Round - Problem 18

Let a parabola $y=x^2$ and a circle $\gamma$ share a common tangent at $(1,1)$. If $\gamma$ is tangent to the positive x-axis and the radius of the circle can be expressed as $\frac{a-\sqrt b}{c}$, where $a,b,c$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b+c$.