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 circles 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 integers, <math> n </math> is not divisible by the square of any prime, and <math> k </math> and <math> p </math> are relatively prime. Find <math> k+m+n+p. </math>
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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>
  
 
== Solution ==
 
== Solution ==
 
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{{solution}}
 
== See also ==
 
== See also ==
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* [[2004 AIME II Problems/Problem 11 | Previous problem]]
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* [[2004 AIME II Problems/Problem 13 | Next problem]]
 
* [[2004 AIME II Problems]]
 
* [[2004 AIME II Problems]]
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[[Category:Intermediate Geometry Problems]]

Revision as of 16:00, 13 October 2006

Problem

Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$and $CD=4.$ Draw circles of radius 3 centered at $A$ and $B,$ and circles of radius 2 centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p,$ where $k, m, n,$ and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p.$

Solution

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See also