Difference between revisions of "2006 AIME I Problems/Problem 8"
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== Solution == | == Solution == | ||
| − | {{ | + | Let <math>x</math> denote the common side length of the rhombi. |
| − | + | Let <math>y</math> denote one of the smaller interior angles of rhombus <math> \mathcal{P} </math>. Then <math>x^2sin(y)=\sqrt{2006}.</math><math> | |
| − | + | \mathcal{T}=x^2\sin(2y) \Rightarrow \mathcal{T}=2x^2siny\cdot cosy \Rightarrow \mathcal{T}= 2\sqrt{2006}\cdot cosy.</math> Thus <math>K</math> is any positive integer on (<math>0, 2\sqrt{2006}</math>). <math>2\sqrt{2006}\approx 89.58</math>. Hence, the number of positive values for <math>K</math> is <math>{089}</math>. | |
== See also == | == See also == | ||
* [[2006 AIME I Problems]] | * [[2006 AIME I Problems]] | ||
[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
Revision as of 18:14, 28 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 
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Solution
Let 
denote the common side length of the rhombi.
Let 
denote one of the smaller interior angles of rhombus 
. Then 
Thus is any positive integer on (
). 
. Hence, the number of positive values for is 
.
