IMO Shortlist 2011 problem G5
Kvaliteta:
Avg: 0,0Težina:
Avg: 8,0 Let
be a triangle with incentre
and circumcircle
. Let
and
be the second intersection points of
with
and
, respectively. The chord
meets
at a point
, and
at a point
. Let
be the intersection point of the line through
parallel to
and the line through
parallel to
. Suppose that the tangents to
at
and
meet at a point
. Prove that the three lines
and
are either parallel or concurrent.
Proposed by Irena Majcen and Kris Stopar, Slovenia
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
![AI](/media/m/d/6/c/d6ca198d9456e8d2c2e0f932598e4a02.png)
![BI](/media/m/3/2/b/32bb635a6f8d93b34190028f9118b7f1.png)
![DE](/media/m/a/c/d/acdf3f4d3c794d9a897484e9d216f5ec.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![BE](/media/m/e/e/2/ee25cd134664bc0c8d7fdbba81e54f90.png)
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
![AE,BD](/media/m/c/e/c/cecef779a7cb44e9cfee00c08ca139d8.png)
![KP](/media/m/7/d/b/7db95e397ea40cc7051be2b300a8102c.png)
Proposed by Irena Majcen and Kris Stopar, Slovenia
Izvor: Međunarodna matematička olimpijada, shortlist 2011