1999 IMO Problems/Problem 1
Problem
Determine all finite sets of at least three points in the plane which satisfy the following condition:
For any two distinct points and in , the perpendicular bisector of the line segment is an axis of symmetry of .
Solution
Upon reading this problem and drawing some points, one quickly realizes that the set consists of all the vertices of any regular polygon.
Now to prove it with some numbers:
Let , with , where is a vertex of a polygon which we can define their coordinates as: for .
That defines the vertices of any regular polygon with being the radius of the circumcircle of the regular -sided polygon.
Now we can pick any points and of the set as:
and , where ; ; and
Then,
and
Let be point which is not part of
Then, , and
The perpendicular bisector of passes through .
Let point , not in be a point that passes through the perpendicular bisector of at a distance from
Then, and
CASE I: is even
and is integer
Then
This means that the perpendicular bisector also passes through a point of
Let be any positive integer
and
Therefore, for any integer .
Also, since for any integer
then this proves that the bisector of any points and is an axis of symmetry for this case.
CASE II: is odd
and is integer
and is integer
Then
This means that the perpendicular bisector does not pass through any point of , but their closest points are and and that
Let be any positive integer
and
Therefore, for any integer .
Since ,
Also, since for any integer
then this proves that the bisector of any points and is an axis of symmetry for this case.
Having proven both cases, then the set of points that comply with the given condition is the set of the vertices of any regular polygon of any number of sides.
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1999 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |