MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2023 Mexico National Olympiad
2
2
Part of
2023 Mexico National Olympiad
Problems
(1)
Numbers on a polygon
Source: Mexico 2023/2
11/8/2023
The numbers from
1
1
1
to
2000
2000
2000
are placed on the vertices of a regular polygon with
2000
2000
2000
sides, one on each vertex, so that the following is true: If four integers
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
satisfy that
1
≤
A
<
B
<
C
<
D
≤
2000
1 \leq A<B<C<D \leq 2000
1
≤
A
<
B
<
C
<
D
≤
2000
, then the segment that joins the vertices of the numbers
A
A
A
and
B
B
B
and the segment that joins the vertices of
C
C
C
and
D
D
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.
combinatorics