Difference between revisions of "2006 IMO Problems/Problem 1"
(→See Also) |
|||
Line 26: | Line 26: | ||
==See Also== | ==See Also== | ||
− | {{IMO box|year=2006| | + | {{IMO box|year=2006|before=First Problem|num-a=2}} |
Latest revision as of 01:02, 19 November 2023
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
and similarly
Since
, we have
It follows that Hence,
and
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 :)
minor edits by lpieleanu
See Also
2006 IMO (Problems) • Resources | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |