IMO Shortlist 1959 problem 5

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2. travnja 2012.
An arbitrary point M is selected in the interior of the segment AB. The square AMCD and MBEF are constructed on the same side of AB, with segments AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and also at another point N. Let N' denote the point of intersection of the straight lines AF and BC.

a) Prove that N and N' coincide;

b) Prove that the straight lines MN pass through a fixed point S independent of the choice of M;

c) Find the locus of the midpoints of the segments PQ as M varies between A and B.
Izvor: Međunarodna matematička olimpijada, shortlist 1959