Difference between revisions of "2005 AIME II Problems/Problem 8"
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== Problem == | == Problem == | ||
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Circles <math> C_1 </math> and <math> C_2 </math> are externally tangent, and they are both internally tangent to circle <math> C_3. </math> The radii of <math> C_1 </math> and <math> C_2 </math> are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of <math> C_3 </math> is also a common external tangent of <math> C_1 </math> and <math> C_2. </math> Given that the length of the chord is <math> \frac{m\sqrt{n}}p </math> where <math> m,n, </math> and <math> p </math> are positive integers, <math> m </math> and <math> p </math> are relatively prime, and <math> n </math> is not divisible by the square of any prime, find <math> m+n+p. </math> | Circles <math> C_1 </math> and <math> C_2 </math> are externally tangent, and they are both internally tangent to circle <math> C_3. </math> The radii of <math> C_1 </math> and <math> C_2 </math> are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of <math> C_3 </math> is also a common external tangent of <math> C_1 </math> and <math> C_2. </math> Given that the length of the chord is <math> \frac{m\sqrt{n}}p </math> where <math> m,n, </math> and <math> p </math> are positive integers, <math> m </math> and <math> p </math> are relatively prime, and <math> n </math> is not divisible by the square of any prime, find <math> m+n+p. </math> | ||
== Solution == | == Solution == | ||
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| + | {{solution}} | ||
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== See Also == | == See Also == | ||
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*[[2005 AIME II Problems]] | *[[2005 AIME II Problems]] | ||
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| + | [[Category:Intermediate Geometry Problems]] | ||
Revision as of 16:50, 7 September 2006
Problem
Circles 
and 
are externally tangent, and they are both internally tangent to circle 
The radii of and are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of 
is also a common external tangent of and 
Given that the length of the chord is 
where 
and 
are positive integers, 
and are relatively prime, and 
is not divisible by the square of any prime, find 
Solution
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