IMO Shortlist 2004 problem G6
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Avg: 8,0 Let be a convex polygon. Prove that there exists a convex hexagon that is contained in and whose area is at least of the area of the polygon .
Alternative version. Let be a convex polygon with vertices. Prove that there exists a convex hexagon with
a) vertices on the sides of the polygon (or)
b) vertices among the vertices of the polygon
such that the area of the hexagon is at least of the area of the polygon.
I couldn't solve this one, partially because I'm not quite sure of the statements' meaning (a) or (b) :)
Obviously if a) is true then so is b).
Alternative version. Let be a convex polygon with vertices. Prove that there exists a convex hexagon with
a) vertices on the sides of the polygon (or)
b) vertices among the vertices of the polygon
such that the area of the hexagon is at least of the area of the polygon.
I couldn't solve this one, partially because I'm not quite sure of the statements' meaning (a) or (b) :)
Obviously if a) is true then so is b).
Izvor: Međunarodna matematička olimpijada, shortlist 2004