1992 IMO Problems/Problem 5
Problem
Let be a finite set of points in three-dimensional space. Let ,,, be the sets consisting of the orthogonal projections of the points of onto the -plane, -plane, -plane, respectively. Prove that
where denotes the number of elements in the finite set . (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane)
Solution
Let be planes with index such that that are parallel to the -plane that contain multiple points of on those planes such that all points of are distributed throughout all planes according to their -coordinates in common.
Let be the number of unique projected points from each to the -plane
Let be the number of unique projected points from each to the -plane
This provides the following:
We also know that
Since be the number of unique projected points from each to the -plane,
if we add them together it will give us the total points projected onto the -plane.
This will be the value of all the elements of
Therefore,
likewise,
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.