IMO Shortlist 2011 problem G7
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Avg: 9,0 Let
be a convex hexagon all of whose sides are tangent to a circle
with centre
. Suppose that the circumcircle of triangle
is concentric with
. Let
be the foot of the perpendicular from
to
. Suppose that the perpendicular from
to
intersects the line
at a point
. Let
be the foot of the perpendicular from
to
. Prove that
.
Proposed by Japan
![ABCDEF](/media/m/9/f/e/9fe205b534135e3a700ffb54d8b96cb0.png)
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
![ACE](/media/m/0/8/3/08350444735a91bb62351789317e7dac.png)
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
![J](/media/m/9/0/e/90ef5cc2558381e341da5808eb92126f.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![DF](/media/m/3/d/d/3dd8b7899102ac0f0d215a5d87897f88.png)
![EO](/media/m/9/e/a/9ea50edb846472f77c9ca1a84cac82bd.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)
![DE](/media/m/a/c/d/acdf3f4d3c794d9a897484e9d216f5ec.png)
![DJ=DL](/media/m/5/b/a/5bae8fc1abe36b151191316c580875ac.png)
Proposed by Japan
Izvor: Međunarodna matematička olimpijada, shortlist 2011