Difference between revisions of "2006 AIME II Problems/Problem 12"
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− | + | == Problem == | |
+ | [[Equilateral triangle | Equilateral]] <math> \triangle ABC </math> is inscribed in a [[circle]] of [[radius]] 2. Extend <math> \overline{AB} </math> through <math> B </math> to point <math> D </math> so that <math> AD=13, </math> and extend <math> \overline{AC} </math> through <math> C </math> to point <math> E </math> so that <math> AE = 11. </math> Through <math> D, </math> draw a line <math> l_1 </math> [[parallel]] to <math> \overline{AE}, </math> and through <math> E, </math> draw a line <math> l_2 </math> parallel to <math> \overline{AD}. </math> Let <math> F </math> be the [[intersection]] of <math> l_1 </math> and <math> l_2. </math> Let <math> G </math> be the point on the circle that is [[collinear]] with <math> A </math> and <math> F </math> and distinct from <math> A. </math> Given that the [[area]] of <math> \triangle CBG </math> can be expressed in the form <math> \frac{p\sqrt{q}}{r}, </math> where <math> p, q, </math> and <math> r </math> are [[positive integer]]s, <math> p </math> and <math> r</math> are [[relatively prime]], and <math> q </math> is not [[divisibility | divisible]] by the [[perfect square | square]] of any [[prime]], find <math> p+q+r. </math> | ||
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+ | == Solution == | ||
+ | {{solution}} | ||
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+ | == See also == | ||
+ | {{AIME box|year=2006|n=II|num-b=11|num-a=13}} | ||
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+ | [[Category:Intermediate Geometry Problems]] |
Revision as of 19:13, 25 September 2007
Problem
Equilateral is inscribed in a circle of radius 2. Extend through to point so that and extend through to point so that Through draw a line parallel to and through draw a line parallel to Let be the intersection of and Let be the point on the circle that is collinear with and and distinct from Given that the area of can be expressed in the form where and are positive integers, and are relatively prime, and is not divisible by the square of any prime, find
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Solution
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See also
2006 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |