IMO Shortlist 2011 problem G1
Dodao/la:
arhiva23. lipnja 2013. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an acute triangle. Let
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
be a circle whose centre
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
lies on the side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. Suppose that
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
is tangent to
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
at
![B'](/media/m/a/1/a/a1a88eb7f35fee4f41c66bfb0c902f51.png)
and
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
at
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
. Suppose also that the circumcentre
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
lies on the shorter arc
![B'C'](/media/m/4/5/8/45835f4a73ab932ee25642d9926e922a.png)
of
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
. Prove that the circumcircle of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
and
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
meet at two points.
Proposed by Härmel Nestra, Estonia
%V0
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
Proposed by Härmel Nestra, Estonia
Izvor: Međunarodna matematička olimpijada, shortlist 2011