Problem A. 380. (October 2005)
A. 380. The convex nsided polygon K lies in the interior of a unit square. Show that it is possible to select three vertices of the polygon that form a triangle of smaller area than units.
(5 pont)
Deadline expired on 15 November 2005.
Solution. We prove that there exist two adjacent sides of K which form a sufficiently small triangle.
K is convex and is a subset of the unit square. Therefore its perimeter p is at most 4.
Denote the length of the ith side of K by d_{i} and let the angle between the ith and (i+1)th sides be _{i}. Since
there exists an index i for which .
Consider the triangle determnined by the ith and (i+1)th sides; its area is
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