IMO Shortlist 2011 problem G4
Kvaliteta:
Avg: 0,0Težina:
Avg: 7,0 Let
be an acute triangle with circumcircle
. Let
be the midpoint of
and let
be the midpoint of
. Let
be the foot of the altitude from
and let
be the centroid of the triangle
. Let
be a circle through
and
that is tangent to the circle
at a point
. Prove that the points
and
are collinear.
Proposed by Ismail Isaev and Mikhail Isaev, Russia
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
![B_0](/media/m/5/a/7/5a7b148f9ae7eef70595a0deebfddd3a.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![C_0](/media/m/e/9/e/e9eb7207b3e27429b1d887f6793224be.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
![B_0](/media/m/5/a/7/5a7b148f9ae7eef70595a0deebfddd3a.png)
![C_0](/media/m/e/9/e/e9eb7207b3e27429b1d887f6793224be.png)
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
![X\not= A](/media/m/d/d/5/dd5220369fe0206ce45c3c1eaa696e19.png)
![D,G](/media/m/4/5/0/450179eafaba7fd6247b0a9361c93a89.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
Proposed by Ismail Isaev and Mikhail Isaev, Russia
Izvor: Međunarodna matematička olimpijada, shortlist 2011
Komentari:
fini_keksi, 25. siječnja 2023. 10:51