2006 IMO Problems/Problem 1
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Problem
Let be triangle with incenter
. A point
in the interior of the triangle satisfies
. Show that
, and that equality holds if and only if
Solution
We have
(1)
and similarly
(2).
Since
, we have
(3).
By (1), (2), and (3), we get ; hence
are concyclic.
Let ray meet the circumcircle of
at point
. Then, by the Incenter-Excenter Lemma,
.
Finally, (since triangle APJ can be degenerate, which happens only when
), but
; hence
and we are done.
By Mengsay LOEM , Cambodia IMO Team 2015
latexed by tluo5458 :)