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ICMC 2019/20 Round 2, Problem 4

Source: Imperial College Mathematics Competition 2019/20 - Round 2

August 7, 2020
college contests

Problem Statement

Let S={S1,S2,…,Sn}\mathcal{S}=\left\{S_1,S_2,\ldots,S_n\right\} be a set of n≥2020n\geq 2020 distinct points on the Euclidean plane, no three of which are collinear. Andy the ant starts at some point Si1S_{i_1} in S\mathcal{S} and wishes to visit a series of 2020 points {Si1,Si2,…,Si2020}⊆S\left\{S_{i_1},S_{i_2},\ldots,S_{i_{2020}}\right\}\subseteq\mathcal{S} in order, such that ij>iki_j>i_k whenever j>kj>k. It is known that ants can only travel between points in S\mathcal{S} in straight lines, and that an ant's path can never self-intersect.
Find a positive integer nn such that Andy can always fulfill his wish.
(Lower n will be awarded more marks. Bounds for this problem may be used as a tie-breaker, should the need to do so arise.)
Proposed by the ICMC Problem Committee
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