Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 10"
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==Solution== | ==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/Problem 9 | Previous Problem]] | + | *[[Mock AIME 1 2006-2007 Problems/Problem 9 | Previous Problem]] |
| − | *[[Mock AIME 1 2006-2007/Problem 11 | Next Problem]] | + | *[[Mock AIME 1 2006-2007 Problems/Problem 11 | Next Problem]] |
*[[Mock AIME 1 2006-2007]] | *[[Mock AIME 1 2006-2007]] | ||
Latest revision as of 22:58, 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 
.
