Difference between revisions of "1992 IMO Problems/Problem 5"
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Revision as of 13:04, 12 November 2023
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
![\[|S|^{2} \le |S_{x}| \cdot |S_{y}| \cdot |S_{z}|,\]](http://latex.artofproblemsolving.com/4/b/8/4b8ce4523020572a5cdf3349b8b11c1a428dce77.png)
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 a plane with index 
such that 
that are parallel to the -plane that contain multiple points
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
