Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 10"

Revision as of 17:27, 19 December 2012

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

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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 $m+n=23$ . File:C:\Documents and Settings\Eigenaar\Mijn documenten\Documenten van Valère\Mock06-07,10.PNG

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Mock AIME 1 2006-2007

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 ==Solution==
 {{solution}}
+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.
+Given <math> \frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n} =\frac{20}{3}</math>, so <math>m+n=23</math>.
+[[File:C:\Documents and Settings\Eigenaar\Mijn documenten\Documenten van Valère\Mock06-07,10.PNG]]
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Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 10"

Revision as of 17:27, 19 December 2012

Problem

Solution