IMO Shortlist 2011 problem G6
Kvaliteta:
Avg: 0,0Težina:
Avg: 8,0 Let
be a triangle with
and let
be the midpoint of
. The angle bisector of
intersects the circle through
and
at the point
inside the triangle
. The line
intersects the circle through
and
in two points
and
. The lines
and
meet at a point
, and the lines
and
meet at a point
. Show that
is the incentre of triangle
.
Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![AB=AC](/media/m/8/7/f/87ff09816431c9a98292e3c223e28c96.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![\angle BAC](/media/m/b/2/1/b21a9e466104c5d33646432221e142be.png)
![D,B](/media/m/6/6/9/6691cc0641b2da217c5cd24b48c8288e.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
![A,E](/media/m/b/7/f/b7f77a58220d143002f3c6756d7b94ba.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![AF](/media/m/a/e/4/ae455e708e936870cb86e6a074a2c5a0.png)
![BE](/media/m/e/e/2/ee25cd134664bc0c8d7fdbba81e54f90.png)
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
![CI](/media/m/b/6/5/b656d89737b7825cdbf637e863735c23.png)
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
![KAB](/media/m/6/5/5/655339e7d2350a3d1e75e4327db72301.png)
Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea
Izvor: Međunarodna matematička olimpijada, shortlist 2011