IMO 2014/P4
Read the "IMO 2014/P4 - addendum" first. We construct G as the intersection point of AH and the circle (O). We construct M' and N' as intersections of lines and our aim is to show that M' = M and N' = N. Using our findings from the "addendum" we have AH = HG, furthermore we can easily show that
angle( GM'A ) = angle( BAG ) = angle( APH ), so PH || M'G, and therefore P is the midpoint of AM', hence M' = M.
Ps. This is supposed to only provide guidance for the proof - fill in the blanks to convince yourself.