Difference between revisions of "2006 IMO Problems/Problem 1"
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Revision as of 00:02, 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 [[2006 IMO Problems/Problem {{{num-b}}}|Problem {{{num-b}}}]] |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |