IMO Shortlist 1999 problem G8
Dodao/la:
arhiva2. travnja 2012. Given a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. The points
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
divide the circumcircle
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
into three arcs
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
. Let
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
be a variable point on the arc
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, and let
![O_{1}](/media/m/a/4/0/a4032d61b59d798b9dac58cad84150cd.png)
and
![O_{2}](/media/m/c/7/0/c7092dc3959c183ed68e2e991c40970e.png)
be the incenters of the triangles
![CAX](/media/m/a/6/8/a680794c7f49bdd85ac223936dd63f8d.png)
and
![CBX](/media/m/a/2/f/a2f8bb9cac600dd9283566e571097337.png)
. Prove that the circumcircle of the triangle
![XO_{1}O_{2}](/media/m/7/f/0/7f0f9c499d77f42c5c8abe540b9143e4.png)
intersects the circle
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
in a fixed point.
%V0
Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.
Izvor: Međunarodna matematička olimpijada, shortlist 1999