2012 AMC 12B Problems/Problem 20
Problem 20
A trapezoid has side lengths 3, 5, 7, and 11. The sums of all the possible areas of the trapezoid can be written in the form of , where
,
, and
are rational numbers and
and
are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to
?
Solution
Name the trapezoid , where
is parallel to
,
, and
. Draw a line through
parallel to
, crossing the side
at
. Then
,
. One needs to guarantee that
, so there are only three possible trapezoids:
In the first case, , so
. Therefore the area of this trapezoid is
.
In the first case, , so
. Therefore the area of this trapezoid is
.
In the first case, , therefore the area of this trapezoid is
.
So , which is rounded down to
.