Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 9"

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== Problem ==
 
== Problem ==
In right triangle <math>\displaystyle ABC,</math> <math>\displaystyle \angle C=90^\circ.</math> Cevians <math>\displaystyle AX</math> and <math>\displaystyle BY</math> intersect at <math>\displaystyle P</math> and are drawn to <math>\displaystyle BC</math> and <math>\displaystyle AC</math> respectively such that <math>\displaystyle \frac{BX}{CX}=\frac23</math> and <math>\displaystyle \frac{AY}{CY}=\sqrt 3.</math> If <math>\displaystyle \tan \angle APB= \frac{a+b\sqrt{c}}{d},</math> where <math>\displaystyle a,b,</math> and <math>\displaystyle d</math> are relatively prime and <math>\displaystyle c</math> has no perfect square divisors excluding <math>\displaystyle 1,</math> find <math>\displaystyle a+b+c+d.</math>
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In right triangle <math>ABC,</math> <math>\angle C=90^\circ.</math> Cevians <math>AX</math> and <math>BY</math> intersect at <math>P</math> and are drawn to <math>BC</math> and <math>AC</math> respectively such that <math>\frac{BX}{CX}=\frac23</math> and <math>\frac{AY}{CY}=\sqrt 3.</math> If <math>\tan \angle APB= \frac{a+b\sqrt{c}}{d},</math> where <math>a,b,</math> and <math>d</math> are relatively prime and <math>c</math> has no perfect square divisors excluding <math>1,</math> find <math>a+b+c+d.</math>
  
 
==Solution==
 
==Solution==
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*[[Mock AIME 2 2006-2007/Problem 8 | Previous Problem]]
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*[[Mock AIME 2 2006-2007 Problems/Problem 8 | Previous Problem]]
  
*[[Mock AIME 2 2006-2007/Problem 10 | Next Problem]]
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*[[Mock AIME 2 2006-2007 Problems/Problem 10 | Next Problem]]
  
 
*[[Mock AIME 2 2006-2007]]
 
*[[Mock AIME 2 2006-2007]]

Revision as of 15:33, 3 April 2012

Problem

In right triangle $ABC,$ $\angle C=90^\circ.$ Cevians $AX$ and $BY$ intersect at $P$ and are drawn to $BC$ and $AC$ respectively such that $\frac{BX}{CX}=\frac23$ and $\frac{AY}{CY}=\sqrt 3.$ If $\tan \angle APB= \frac{a+b\sqrt{c}}{d},$ where $a,b,$ and $d$ are relatively prime and $c$ has no perfect square divisors excluding $1,$ find $a+b+c+d.$

Solution

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