Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 10"
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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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+ | ==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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+ | ---- | ||
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+ | *[[Mock AIME 1 2006-2007 Problems/Problem 9 | Previous Problem]] | ||
+ | |||
+ | *[[Mock AIME 1 2006-2007 Problems/Problem 11 | Next Problem]] | ||
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+ | *[[Mock AIME 1 2006-2007]] |
Latest revision as of 23:58, 24 April 2013
Problem
In ,
,
, and
have lengths
,
, and
, respectively. Let the incircle, circle
, of
touch
,
, and
at
,
, and
, respectively. Construct three circles,
,
, and
, externally tangent to the other two and circles
,
, and
are internally tangent to the circle
at
,
, and
, respectively. Let circles
,
,
, and
have radii
,
,
, and
, respectively. If
where
and
are positive integers, find
.
Solution
Radius , radius
, radius
and
, see picture.
Given , so
.