Difference between revisions of "2006 AIME I Problems/Problem 8"
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== Problem == | == Problem == | ||
− | Hexagon <math> ABCDEF </math> is divided into four | + | [[Hexagon]] <math> ABCDEF </math> is divided into four [[rhombus]]es, <math> \mathcal{P, Q, R, S,} </math> and <math> \mathcal{T,} </math> as shown. Rhombuses <math> \mathcal{P, Q, R,} </math> and <math> \mathcal{S} </math> are [[congruent (geometry) | congruent]], and each has [[area]] <math> \sqrt{2006}. </math> Let <math> K </math> be the area of rhombus <math> \mathcal{T}</math>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</math>. |
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== Solution == | == Solution == | ||
Let <math>x</math> denote the common side length of the rhombi. | Let <math>x</math> denote the common side length of the rhombi. | ||
− | Let <math>y</math> denote one of the smaller interior | + | Let <math>y</math> denote one of the smaller interior [[angle]]s of rhombus <math> \mathcal{P} </math>. Then <math>x^2\sin(y)=\sqrt{2006}</math>. We also see that <math>\displaystyle K=x^2\sin(2y) \Longrightarrow K=2x^2\sin y \cdot \cos y \Longrightarrow K = 2\sqrt{2006}\cdot \cos y</math>. Thus <math>K</math> can be any positive integer in the [[interval]] <math>(0, 2\sqrt{2006})</math>. |
− | + | <math>2\sqrt{2006} = \sqrt{8024}</math> and <math>89^2 = 7921 < 8024 < 8100 = 90^2</math>, so <math>K</math> can be any [[integer]] between 1 and 89, inclusive. Thus the number of positive values for <math>K</math> is 089. | |
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== See also == | == See also == | ||
+ | * [[2006 AIME I Problems/Problem 7 | Previous problem]] | ||
+ | * [[2006 AIME I Problems/Problem 9 | Next problem]] | ||
* [[2006 AIME I Problems]] | * [[2006 AIME I Problems]] | ||
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
+ | [[Category:Intermediate Trigonometry Problems]] |
Revision as of 12:34, 29 November 2006
Problem
Hexagon is divided into four rhombuses,
and
as shown. Rhombuses
and
are congruent, and each has area
Let
be the area of rhombus
. Given that
is a positive integer, find the number of possible values for
.
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Solution
Let denote the common side length of the rhombi.
Let
denote one of the smaller interior angles of rhombus
. Then
. We also see that
. Thus
can be any positive integer in the interval
.
and
, so
can be any integer between 1 and 89, inclusive. Thus the number of positive values for
is 089.