IMO Shortlist 1993 problem G2
Kvaliteta:
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Avg: 6,0 A circle bisects a circle if it cuts at opposite ends of a diameter. , , are circles with distinct centers (respectively).
Show that are collinear iff there is no unique circle which bisects each of , , . Show that if there is more than one circle which bisects each of , , , then all such circles pass through two fixed points. Find these points.
Original Statement:
A circle is said to cut a circle diametrically if and only if their common chord is a diameter of
Let be three circles with distinct centres respectively. Prove that are collinear if and only if there is no unique circle which cuts each of diametrically. Prove further that if there exists more than one circle which cuts each diametrically, then all such circles pass through two fixed points. Locate these points in relation to the circles
Show that are collinear iff there is no unique circle which bisects each of , , . Show that if there is more than one circle which bisects each of , , , then all such circles pass through two fixed points. Find these points.
Original Statement:
A circle is said to cut a circle diametrically if and only if their common chord is a diameter of
Let be three circles with distinct centres respectively. Prove that are collinear if and only if there is no unique circle which cuts each of diametrically. Prove further that if there exists more than one circle which cuts each diametrically, then all such circles pass through two fixed points. Locate these points in relation to the circles
Izvor: Međunarodna matematička olimpijada, shortlist 1993