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Dirichlet strikes back

Source: AIMO 2007, TST 2, P2

January 11, 2009
inductioncombinatorics unsolvedcombinatorics

Problem Statement

Let n,k∈N n, k \in \mathbb{N} with 1≤k≤n2−1. 1 \leq k \leq \frac {n}{2} - 1. There are n n points given on a circle. Arbitrarily we select nk+1 nk + 1 chords among the points on the circle. Prove that of these chords there are at least k+1 k + 1 chords which pairwise do not have a point in common.
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