MathDB

Let

S

be a unit circle and

K

a subset of

S

consisting of several closed arcs. Let

K

satisfy the following properties:

(\text{i})

K

contains three points

A,B,C

, that are the vertices of an acute-angled triangle

(\text{ii})

For every point

A

that belongs to

K

its diametrically opposite point

A'

and all points

B

on an arc of length

\frac{1}{9}

with center

A'

do not belong to

K

. Prove that there are three points

E,F,G

S

that are vertices of an equilateral triangle and that do not belong to

K

Problem Statement