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Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 14"

 
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==Problem==
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Circles <math>\omega_1</math> and <math>\omega_2</math> are centered on opposite sides of line <math>l</math>, and are both tangent to <math>l</math> at <math>P</math>. <math>\omega_3</math> passes through <math>P</math>, intersecting <math>l</math> again at <math>Q</math>. Let <math>A</math> and <math>B</math> be the intersections of <math>\omega_1</math> and <math>\omega_3</math>, and <math>\omega_2</math> and <math>\omega_3</math> respectively. <math>AP</math> and <math>BP</math> are extended past <math>P</math> and intersect <math>\omega_2</math> and <math>\omega_1</math> at <math>C</math> and <math>D</math> respectively. If <math>AD = 3, AP = 6, DP = 4,</math> and <math>PQ = 32</math>, then the area of triangle <math>PBC</math> can be expressed as <math>\frac{p\sqrt{q}}{r}</math>, where <math>p, q,</math> and <math>r</math> are positive integers such that <math>p</math> and <math>r</math> are coprime and <math>q</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>.
  
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<math>14.</math> Circles <math>\omega_1</math> and <math>\omega_2</math> are centered on opposite sides of line <math>l</math>, and are both tangent to <math>l</math> at <math>P</math>. <math>\omega_3</math> passes through <math>P</math>, intersecting <math>l</math> again at <math>Q</math>. Let <math>A</math> and <math>B</math> be the intersections of <math>\omega_1</math> and <math>\omega_3</math>, and <math>\omega_2</math> and <math>\omega_3</math> respectively. <math>AP</math> and <math>BP</math> are extended past <math>P</math> and intersect <math>\omega_2</math> and <math>\omega_1</math> at <math>C</math> and <math>D</math> respectively. If <math>AD = 3, AP = 6, DP = 4,</math> and <math>PQ = 32</math>, then the area of triangle <math>PBC</math> can be expressed as <math>\frac{p\sqrt{q}}{r}</math>, where <math>p, q,</math> and <math>r</math> are positive integers such that <math>p</math> and <math>r</math> are coprime and <math>q</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>.
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==Solution==
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{{solution}}
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==See also==

Revision as of 07:30, 14 February 2008

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

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See also

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