IMO Shortlist 1988 problem 18

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Dodao/la: arhiva
2. travnja 2012.
Consider 2 concentric circle radii R and r (R > r) with centre O. Fix P on the small circle and consider the variable chord PA of the small circle. Points B and C lie on the large circle; B,P,C are collinear and BC is perpendicular to AP.

i.) For which values of \angle OPA is the sum BC^2 + CA^2 + AB^2 extremal?

ii.) What are the possible positions of the midpoints U of BA and V of AC as \angle OPA varies?
Izvor: Međunarodna matematička olimpijada, shortlist 1988