IMO Shortlist 2006 problem G7
Kvaliteta:
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Avg: 9,0 In a triangle , let , , be the midpoints of the sides , , , respectively, and , , be the midpoints of the arcs , , of the circumcircle of , not containing the vertices , , , respectively. For , let be the circle with as diameter. Let be the common external common tangent to the circles and (for all ) such that lies on the opposite side of than and do.
Prove that the lines , , form a triangle similar to and find the ratio of similitude.
Prove that the lines , , form a triangle similar to and find the ratio of similitude.
Izvor: Međunarodna matematička olimpijada, shortlist 2006