Mock AIME 6 2006-2007 Problems/Problem 6

Problem

$C_1$ is a circle with radius $164$ and $C_2$ is a circle internally tangent to $C_1$ that passes through the center of $C_1$ . $\overline{AB}$ is a chord in $C_1$ of length $320$ tangent to $C_2$ at $D$ where $AD>BD$ . Given that $BD=a-b\sqrt{c}$ where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime, what is $a+b+c$ ?

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Mock AIME 6 2006-2007 Problems/Problem 6

Problem

Solution