IMO Shortlist 2009 problem G2
Kvaliteta:
Avg: 3,5Težina:
Avg: 6,0 Let
be a triangle with circumcentre
. The points
and
are interior points of the sides
and
respectively. Let
and
be the midpoints of the segments
and
. respectively, and let
be the circle passing through
and
. Suppose that the line
is tangent to the circle
. Prove that
Proposed by Sergei Berlov, Russia
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![K,L](/media/m/c/f/8/cf8b7b8c56970a06671ff82ddb7f6450.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![BP,CQ](/media/m/6/2/f/62f9112941d60283b09cdc1c5b2874ad.png)
![PQ](/media/m/f/2/f/f2f65ec376294df7eca22d2c1a189747.png)
![\Gamma](/media/m/4/e/0/4e08987e1d0700578a2eb5c2fc65dc3b.png)
![K,L](/media/m/c/f/8/cf8b7b8c56970a06671ff82ddb7f6450.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![PQ](/media/m/f/2/f/f2f65ec376294df7eca22d2c1a189747.png)
![\Gamma](/media/m/4/e/0/4e08987e1d0700578a2eb5c2fc65dc3b.png)
![OP = OQ.](/media/m/c/0/1/c0135f985d771c12d363a4df1deeb689.png)
Proposed by Sergei Berlov, Russia
Izvor: Međunarodna matematička olimpijada, shortlist 2009