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,
We also know that the total number of elements of each is less or equal to the total number of elements in
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.