Difference between revisions of "1992 IMO Problems/Problem 5"
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likewise, | likewise, | ||
− | <math>|S_{y}|=\sum_{i=1}^{n} | + | <math>|S_{y}|=\sum_{i=1}^{n}b_{i}</math> |
Revision as of 14:19, 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:
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,
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.