Mock AIME 3 Pre 2005 Problems/Problem 14

Problem

Circles $\omega_1$ and $\omega_2$ are centered on opposite sides of line $l$ , and are both tangent to $l$ at $P$ . $\omega_3$ passes through $P$ , intersecting $l$ again at $Q$ . Let $A$ and $B$ be the intersections of $\omega_1$ and $\omega_3$ , and $\omega_2$ and $\omega_3$ respectively. $AP$ and $BP$ are extended past $P$ and intersect $\omega_2$ and $\omega_1$ at $C$ and $D$ respectively. If $AD = 3, AP = 6, DP = 4,$ and $PQ = 32$ , then the area of triangle $PBC$ can be expressed as $\frac{p\sqrt{q}}{r}$ , where $p, q,$ and $r$ are positive integers such that $p$ and $r$ are coprime and $q$ is not divisible by the square of any prime. Determine $p + q + r$ .

Solution

Let $O_1, O_2,$ and $O_3$ be the centers of $\omega_1, \omega_2$ and $\omega_3$ respectively.

Let point $R$ be the midpoint of $QP$ . Thus, $O_3R \bot PQ$ and $|PR|=\frac{|PR|}{2}=16$

Invert about a circle with radius 1 and center P. Note that since all relevant circles and lines go through P, they all are transformed into lines, and $\omega_1,\omega_2, l$ are all tangent at infinity (i.e. parallel). That was the crux move; some more basic length chasing using similar triangles gets you the answer.

~Tomas Diaz. orders@tomasdiaz.com

Mock AIME 3 Pre 2005 (Problems, Source)
Preceded by Problem 13	Followed by Problem 15
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Mock AIME 3 Pre 2005 Problems/Problem 14

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