Olympiad Geometry Problem #86: IMO Shortlist 2019 G2

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Here is a fascinating problem that was recently released as one of the candidate problems for the 2019 International Math Olympiad (although it didn't end up making it on the exam). The solution I give is not the simplest, but is fairly insightful in my opinion. Enjoy! Links below.
  1. Michael Greenberg
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truely amazing! thanks for uploading beautiful solution.

김동욱-wtz
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Nice solution👌🏻
I solved the problem in the same way but I did the step, showing that RO and KL are perpendicular, in another (I think shorter) way. Because K, O, L and R are the Midpoints of the quadrilateral IJNM, KOLR has to be a parallelogram. Because of DJ=DN, DM=DI and <JDM=<NDI, we know, JDM~NDI an therefor JM=NI. Thus KOLR is a Rhombus and so the diagonals KL and RO are perpendicular.

germanyop
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When the gilding principle is used I still don't get what is the definition of oriented the same way.

helo