Difference between revisions of "2006 AIME I Problems/Problem 8"
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== See also == | == See also == | ||
+ | * [[2006 AIME I Problems]] | ||
* [[2006 AIME I Problems/Problem 7 | Previous problem]] | * [[2006 AIME I Problems/Problem 7 | Previous problem]] | ||
* [[2006 AIME I Problems/Problem 9 | Next problem]] | * [[2006 AIME I Problems/Problem 9 | Next problem]] | ||
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
[[Category:Intermediate Trigonometry Problems]] | [[Category:Intermediate Trigonometry Problems]] |
Revision as of 20:50, 11 March 2007
Problem
Hexagon is divided into five 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
.
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.