2006 AIME I Problems/Problem 8
Revision as of 20:22, 29 November 2006 by Xantos C. Guin (talk | contribs) (there are 5 not 4 rhombuses according to the original problem.)
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
.
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
. 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.