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Difference between revisions of "2015 AIME II Problems/Problem 15"

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==Problem==
 
==Problem==
  
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Circles <math>\mathcal{P}</math> and <math>\mathcal{Q}</math> have radii <math>1</math> and <math>4</math>, respectively, and are externally tangent at point <math>A</math>. Point <math>B</math> is on <math>\mathcal{P}</math> and point <math>C</math> is on <math>\mathcal{Q}</math> such that <math>BC</math> is a common external tangent of the two circles. A line <math>\ell</math> through <math>A</math> intersects <math>\mathcal{P}</math> again at <math>D</math> and intersects <math>\mathcal{Q}</math> again at <math>E</math>. Points <math>B</math> and <math>C</math> lie on the same side of <math>\ell</math>, and the areas of <math>\triangle DBA</math> and <math>\triangle ACE</math> are equal. This common area is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
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Circles <math>\mathcal{P}</math> and <math>\mathcal{Q}</math> have radii <math>1</math> and <math>4</math>, respectively, and are externally tangent at point <math>A</math>. Point <math>B</math> is on <math>\mathcal{P}</math> and point <math>C</math> is on <math>\mathcal{Q}</math> so that line <math>BC</math> is a common external tangent of the two circles. A line <math>\ell</math> through <math>A</math> intersects <math>\mathcal{P}</math> again at <math>D</math> and intersects <math>\mathcal{Q}</math> again at <math>E</math>. Points <math>B</math> and <math>C</math> lie on the same side of <math>\ell</math>, and the areas of <math>\triangle DBA</math> and <math>\triangle ACE</math> are equal. This common area is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
==Solution==
 
==Solution==

Revision as of 10:33, 27 March 2015

Problem

Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

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