Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 7"
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Revision as of 15:49, 3 April 2012
Problem
Let have
and
. Point
is such that
and
. Construct point
on segment
such that
.
and
are extended to meet at
. If
where
and
are positive integers, find
(note:
denotes the area of
).
Solution
We can immediately see that quadrilateral is cyclic, since
. We then have, from Power of a Point, that
. In other words,
.
is then 2, and
is 1. We can now use Menelaus on line
with respect to triangle
:
This shows that .
Now let , for some real
. Therefore
, and
. Similarly,
and
. The desired ratio is then
Therefore .