9

Part of 2009 Sharygin Geometry Olympiad

Problems(1)

n points on the plane

Source: I.F.Sharygin contest 2009 - Correspondence round - Problem 9

n

points on the plane, which are the vertices of a convex polygon,

n > 3

k

regular triangles with the side equal to

1

and the vertices at the given points. [*] Prove that

k < \frac {2}{3}n

. [*] Construct the configuration with

k > 0.666n

rotationinequalitiesgeometry proposedgeometry