IMO 1979 LL YUG79

Origin: YUG

Problem

Let S be a unit circle and K a subset of S consisting of several closed arcs. Let K satisfy the following properties: (i) K contains three points A, B, C, that are the vertices of an acute- angled triangle; (ii) for every point A that belongs to K its diametrically opposite point A′ and all points B on an arc of length 1/9 with center A′ do not belong to K. Prove that there are three points E, F, G on S that are vertices of an equilateral triangle and that do not belong to K.