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 17:45, 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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