1973 IMO Problems/Problem 2
Problem
Determine whether or not there exists a finite set of points in space not lying in the same plane such that, for any two points and of ; one can select two other points and of so that lines and are parallel and not coincident.
Solution
In order to solve this problem we can start by finding at least one finite set that satisfies the condition.
We start by defining our first set with the vertices of a cube of side as follows:
Since all the faces of this cube have a parallel face, then any two points on one face will have corresponding 2 points on the opposite face that is parallel. However we have four diagonals on this cube that do not have two points that are parallel to any of these diagonals.
By doing a reflection of the points on the plane along the these four diagonals will have their respective parallel diagonals on the $x \le 0& space
If set$ (Error compiling LaTeX. Unknown error_msg)M$of points in space consist of 3 points or less, then we can't satisfy the condition because we would need at least 4 points.
If set$ (Error compiling LaTeX. Unknown error_msg)MABCDMM$with 4 points would not satisfy the condition.
If set$ (Error compiling LaTeX. Unknown error_msg)MMABCDABCD$parallel because any of those combinations of lines will be skewed. be parallel because those other two points will provide skew lines.
If set$ (Error compiling LaTeX. Unknown error_msg)MABCDEFABCDEFAE$, such diagonal can't be parallel with anything else.
If set$ (Error compiling LaTeX. Unknown error_msg)MABCDEFGMAGHIAH$not parallel to anything else. ...and you keep adding points until infinity at which time the condition will be satisfied. But that would make the set infinite and not finite.
Therefore the finite set$ (Error compiling LaTeX. Unknown error_msg)M$ of points in space for this problem does not exist.
NOTE: I made a mistake. I found a case that it does exist. I'm working on a solution to update this.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1973 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |