Difference between revisions of "2004 AIME II Problems/Problem 12"
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== Problem == | == Problem == | ||
| − | Let <math> ABCD </math> be an [[isosceles trapezoid]], whose dimensions are <math> AB = 6, BC=5=DA, </math>and <math> CD=4. </math> Draw [[circle]]s of [[radius]] 3 centered at <math> A </math> and <math> B, </math> and circles of radius 2 centered at <math> C </math> and <math> D. </math> A circle contained within the trapezoid is [[tangent]] to all four of these circles. Its radius is <math> \frac{-k+m\sqrt{n}}p, </math> where <math> k, m, n, </math> and <math> p </math> are [[positive integer]]s, <math> n </math> is not [[divisibility | divisible]] by the [[square]] of any [[prime number | prime]], and <math> k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> | + | Let <math> ABCD </math> be an [[isosceles trapezoid]], whose dimensions are <math> AB = 6, BC=5=DA, </math>and <math> CD=4. </math> Draw [[circle]]s of [[radius]] 3 centered at <math> A </math> and <math> B, </math> and circles of radius 2 centered at <math> C </math> and <math> D. </math> A circle contained within the trapezoid is [[tangent]] to all four of these circles. Its radius is <math> \frac{-k+m\sqrt{n}}p, </math> where <math> k, m, n, </math> and <math> p </math> are [[positive integer]]s, <math> n </math> is not [[divisibility | divisible]] by the [[perfect square | square]] of any [[prime number | prime]], and <math> k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> |
== Solution == | == Solution == | ||
Revision as of 16:45, 17 October 2006
Problem
Let 
be an isosceles trapezoid, whose dimensions are 
and 
Draw circles of radius 3 centered at 
and 
and circles of radius 2 centered at 
and 
A circle contained within the trapezoid is tangent to all four of these circles. Its radius is 
where 
and 
are positive integers, 
is not divisible by the square of any prime, and 
and are relatively prime. Find 
Solution
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