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

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. This problem needs a solution. If you have a solution for it, please help us out by adding it.

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Mock AIME 3 Pre 2005 Problems/Problem 14

Problem

Solution

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