IMO Shortlist 2005 problem G6
Kvaliteta:
Avg: 0,0Težina:
Avg: 8,0 Let
be a triangle, and
the midpoint of its side
. Let
be the incircle of triangle
. The median
of triangle
intersects the incircle
at two points
and
. Let the lines passing through
and
, parallel to
, intersect the incircle
again in two points
and
. Let the lines
and
intersect
again at the points
and
. Prove that
.
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![\gamma](/media/m/2/4/a/24aca7af13a8211060a900a49ef999e9.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![\gamma](/media/m/2/4/a/24aca7af13a8211060a900a49ef999e9.png)
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![\gamma](/media/m/2/4/a/24aca7af13a8211060a900a49ef999e9.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
![AX](/media/m/3/a/8/3a8b3cfe621304b5621fb712075419c2.png)
![AY](/media/m/b/b/9/bb90496e08c85f2dceaaf0a186d021fe.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
![BP = CQ](/media/m/f/b/9/fb9909fd1cf9ceb5780c25a9187dc6e0.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2005