IMO Shortlist 1992 problem 10

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Dodao/la: arhiva
April 2, 2012
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 \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, where \vert A \vert denotes the number of elements in the finite set A.

Note Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.
Source: Međunarodna matematička olimpijada, shortlist 1992