MEMO 2017 ekipno problem 6

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Let ABC be an acute-angled triangle with AB \neq AC, circumcentre O and circumcircle \Gamma. Let the tangents to \Gamma at B and C meet each other at D, and let the line AO intersect BC at E. Denote the midpoint of BC by M and let AM meet \Gamma again at N \neq A. Finally, let F \neq A be a point on \Gamma such that A, M, E and F are concyclic.
Prove that FN bisects the segment MD.

Source: Srednjoeuropska matematička olimpijada 2017, ekipno natjecanje, problem 6