2007 BMO Problems/Problem 1
Revision as of 22:19, 2 May 2007 by Boy Soprano II (talk | contribs)
Problem
(Albania)
Let be a convex quadrilateral with
and
not equal to
, and let
be the intersection point of its diagonals. Prove that
if and only if
.
Solution
Since ,
, and similarly,
. Since
, by consdering triangles
we have
. It follows that
.
Now, by the Law of Sines,
.
It follows that if and only if
.
Since ,

and

From these inequalities, we see that if and only if
(i.e.,
) or
(i.e.,
). But if
, then triangles
are congruent and
, a contradiction. Thus we conclude that
if and only if
, Q.E.D.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.