MathDB

The numbers from

1

2000

are placed on the vertices of a regular polygon with

2000

sides, one on each vertex, so that the following is true: If four integers

A, B, C, D

satisfy that

1 \leq A<B<C<D \leq 2000

, then the segment that joins the vertices of the numbers

A

and

B

and the segment that joins the vertices of

C

and

D

do not intersect inside the polygon. Prove that there exists a perfect square such that the number diametrically opposite to it is not a perfect square.

Problem Statement