IMO Shortlist 1993 problem G2


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2. travnja 2012.
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A circle S bisects a circle S' if it cuts S' at opposite ends of a diameter. S_A, S_B,S_C are circles with distinct centers A, B, C (respectively).
Show that A, B, C are collinear iff there is no unique circle S which bisects each of S_A, S_B,S_C . Show that if there is more than one circle S which bisects each of S_A, S_B,S_C , then all such circles pass through two fixed points. Find these points.

Original Statement:

A circle S is said to cut a circle \Sigma diametrically if and only if their common chord is a diameter of \Sigma.
Let S_A, S_B, S_C be three circles with distinct centres A,B,C respectively. Prove that A,B,C are collinear if and only if there is no unique circle S which cuts each of S_A, S_B, S_C diametrically. Prove further that if there exists more than one circle S which cuts each S_A, S_B, S_C diametrically, then all such circles S pass through two fixed points. Locate these points in relation to the circles S_A, S_B, S_C.
Izvor: Međunarodna matematička olimpijada, shortlist 1993