Difference between revisions of "1973 IMO Problems/Problem 2"
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If set <math>M</math> 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 <math>M</math> 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 <math>M</math> of points in space consist of 4 points, then we can't satisfy the condition because for the condition of lines <math>AB</math> and <math>CD</math> to be parallel the 4 points would need to be co-planar. But the points in set <math>M</math> shall not be lying in the same plane. | + | If set <math>M</math> of points in space consist of 4 points, then we can't satisfy the condition because for the condition of lines <math>AB</math> and <math>CD</math> to be parallel the 4 points would need to be co-planar. But the points in set <math>M</math> shall not be lying in the same plane. |
+ | So, a finite set <math>M</math> with 4 points would not satisfy the condition. | ||
If set <math>M</math> of points in space consist of 5 points, then we can't satisfy the condition either because even though we can construct a parallelogram in space with 4 co-planar vertices in the set <math>M</math> with a 5th point outside of the parallelogram plane, the condition is to select any two points. So if one of the <math>A& or </math>B<math> points is that 5th point, there would be no other two other points </math>C<math> and </math>D<math> for which will make lines </math>AB<math> and </math>CD<math> parallel because any of those combinations of lines will be skewed. | If set <math>M</math> of points in space consist of 5 points, then we can't satisfy the condition either because even though we can construct a parallelogram in space with 4 co-planar vertices in the set <math>M</math> with a 5th point outside of the parallelogram plane, the condition is to select any two points. So if one of the <math>A& or </math>B<math> points is that 5th point, there would be no other two other points </math>C<math> and </math>D<math> for which will make lines </math>AB<math> and </math>CD<math> parallel because any of those combinations of lines will be skewed. |
Revision as of 20:50, 11 September 2023
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
If set 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 of points in space consist of 4 points, then we can't satisfy the condition because for the condition of lines
and
to be parallel the 4 points would need to be co-planar. But the points in set
shall not be lying in the same plane.
So, a finite set
with 4 points would not satisfy the condition.
If set of points in space consist of 5 points, then we can't satisfy the condition either because even though we can construct a parallelogram in space with 4 co-planar vertices in the set
with a 5th point outside of the parallelogram plane, the condition is to select any two points. So if one of the $A& or$ (Error compiling LaTeX. Unknown error_msg)B
C
D
AB
CD$parallel because any of those combinations of lines will be skewed.
be parallel because those other two points will provide skew lines.
If set$ (Error compiling LaTeX. Unknown error_msg)MABCDEF
ABC
DEF
AE$, such diagonal can't be parallel with anything else.
If set$ (Error compiling LaTeX. Unknown error_msg)MABCDEFG
M
AG
H% and
parallel to the diagonals, but then the two new points have the lines
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 of points in space for this problem does not exist.
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 |