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Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 10"
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10. In <math>\triangle ABC</math>, <math>AB</math>, <math>BC</math>, and <math>CA</math> have lengths <math>3</math>, <math>4</math>, and <math>5</math>, respectively. Let the incircle, circle <math>I</math>, of <math>\triangle ABC</math> touch <math>AB</math>, <math>BC</math>, and <math>CA</math> at <math>C'</math>, <math>A'</math>, and <math>B'</math>, respectively. Construct three circles, <math>A''</math>, <math>B''</math>, and <math>C''</math>, externally tangent to the other two and circles <math>A''</math>, <math>B''</math>, and <math>C''</math> are internally tangent to the circle <math>I</math> at <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Let circles <math>A''</math>, <math>B''</math>, <math>C''</math>, and <math>I</math> have radii <math>a</math>, <math>b</math>, <math>c</math>, and <math>r</math>, respectively. If <math>\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math>.
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==Problem==
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In <math>\triangle ABC</math>, <math>AB</math>, <math>BC</math>, and <math>CA</math> have lengths <math>3</math>, <math>4</math>, and <math>5</math>, respectively. Let the incircle, circle <math>I</math>, of <math>\triangle ABC</math> touch <math>AB</math>, <math>BC</math>, and <math>CA</math> at <math>C'</math>, <math>A'</math>, and <math>B'</math>, respectively. Construct three circles, <math>A''</math>, <math>B''</math>, and <math>C''</math>, externally tangent to the other two and circles <math>A''</math>, <math>B''</math>, and <math>C''</math> are internally tangent to the circle <math>I</math> at <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Let circles <math>A''</math>, <math>B''</math>, <math>C''</math>, and <math>I</math> have radii <math>a</math>, <math>b</math>, <math>c</math>, and <math>r</math>, respectively. If <math>\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math>.
  
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[[Mock AIME 1 2006-2007]]
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==Solution==
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Radius <math>a=\frac{3}{7}</math>, radius <math>b=\frac{6}{11}</math>, radius <math>c=\frac{2}{5}</math> and <math>r=1</math>, see picture.
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Given <math> \frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n} =\frac{20}{3}</math>, so <math>m+n=23</math>.
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*[[Mock AIME 1 2006-2007 Problems/Problem 9 | Previous Problem]]
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*[[Mock AIME 1 2006-2007 Problems/Problem 11 | Next Problem]]
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*[[Mock AIME 1 2006-2007]]

Latest revision as of 22:58, 24 April 2013

Problem

In $\triangle ABC$, $AB$, $BC$, and $CA$ have lengths $3$, $4$, and $5$, respectively. Let the incircle, circle $I$, of $\triangle ABC$ touch $AB$, $BC$, and $CA$ at $C'$, $A'$, and $B'$, respectively. Construct three circles, $A''$, $B''$, and $C''$, externally tangent to the other two and circles $A''$, $B''$, and $C''$ are internally tangent to the circle $I$ at $A'$, $B'$, and $C'$, respectively. Let circles $A''$, $B''$, $C''$, and $I$ have radii $a$, $b$, $c$, and $r$, respectively. If $\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}$ where $m$ and $n$ are positive integers, find $m+n$.

Solution

Radius $a=\frac{3}{7}$, radius $b=\frac{6}{11}$, radius $c=\frac{2}{5}$ and $r=1$, see picture.

Given $\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n} =\frac{20}{3}$, so $m+n=23$.

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