1992 IMO Problems/Problem 5

Problem

Let $S$ be a finite set of points in three-dimensional space. Let $S_{x}$ , $S_{y}$ , $S_{z}$ , be the sets consisting of the orthogonal projections of the points of $S$ onto the $yz$ -plane, $zx$ -plane, $xy$ -plane, respectively. Prove that

$|S|^{2} \le |S_{x}| \cdot |S_{y}| \cdot |S_{z}|,$

where $|A|$ denotes the number of elements in the finite set $|A|$ . (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane)

Solution

Let $Z_{i}$ be planes with index $i$ such that $1 \le i \le n$ that are parallel to the $xy$ -plane that contain multiple points of $S$ on those planes such that all points of $S$ are distributed throughout all planes $Z_{i}$ according to their $z$ -coordinates in common.