IMO 2014 problem 4

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Dodao/la: arhiva
21. rujna 2014.
Points P and Q lie on side BC of acute-angled triangle ABC so that \angle PAB = \angle BCA and \angle CAQ = \angle ABC. Points M and N lie on lines AP and AQ, respectively, such that P is the midpoint of AM, and Q is the midpoint of AN. Prove that lines BM and CN intersect on the circumcircle of triangle ABC.
Izvor: International Mathematical Olympiad 2014, day 2