2013 Mock AIME I Problems/Problem 1

Problem 1

Two circles $C_1$ and $C_2$ , each of unit radius, have centers $A_1$ and $A_2$ such that $A_1A_2=6$ . Let $P$ be the midpoint of $A_1A_2$ and let $C_3$ be a circle externally tangent to both $C_1$ and $C_2$ . $C_1$ and $C_3$ have a common tangent that passes through $P$ . If this tangent is also a common tangent to $C_2$ and $C_1$ , find the radius of circle $C_3$ .

Solution

Let $A_3$ be the center of circle $C_3$ and $Q$ be the point of tangency between $C_3$ and $C_2$ . Note that triangles $PQA_2$ and $A_3PA_2$ are similar, so $\frac{A_3A_2}{PA_2}=\frac{PA_2}{A_2Q}=3$ and $A_3A_2=r+1=9$ . Thus the radius of $C_3$ is $\boxed{8}$ .

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2013 Mock AIME I Problems/Problem 1

Problem 1

Solution