Difference between revisions of "1992 IMO Problems/Problem 5"
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Substituting [Equation 3] and [Equation 4] into [Equation 7] we get: | Substituting [Equation 3] and [Equation 4] into [Equation 7] we get: | ||
− | <math>|S|^{2} \le |S_{x}|</math> | + | <math>|S|^{2} \le |S_{x}| \cdot |S_{y}| \cdot |S_{z}|</math> |
Revision as of 14:45, 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
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:
[Equation 1]
We also know that
[Equation 2]
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.
Therefore,
[Equation 3]
likewise,
[Equation 4]
We also know that the total number of elements of each is less or equal to the total number of elements in
That is,
[Equation 5]
Multiplying [Equation 1] by [Equation 5] we get:
Therefore,
Adding all we get:
[Equation 6]
Substituting [Equation 2] into [Equation 6] we get:
Since, ,
Then,
[Equation 7]
Substituting [Equation 3] and [Equation 4] into [Equation 7] we get:
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.