MathDB

Let

S

be a finite set of points on a plane, where no three points are collinear, and the convex hull of

S

\Omega

, is a

2016-

gon

A_1A_2\ldots A_{2016}

. Every point on

S

is labelled one of the four numbers

\pm 1,\pm 2

, such that for

i=1,2,\ldots , 1008,

the numbers labelled on points

A_i

and

A_{i+1008}

are the negative of each other. Draw triangles whose vertices are in

S

, such that any two triangles do not have any common interior points, and the union of these triangles is

\Omega

. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

Problem Statement