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Indija mo 2011

India 2011 counting and probability

Problem

Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
Solution
Join the vertices of the nine-sided regular polygon. We get line segments. All these fall into sets of parallel lines. Now using any points, we get line segments. By the pigeon-hole principle, two of these must be parallel. But, these parallel lines determine a trapezium.

Techniques

Pigeonhole principleInscribed/circumscribed quadrilaterals