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Vectors lying inside disc

Source: Simon Marais Mathematics Competition 2023 Paper B Problem 1

October 16, 2023
vectorgeometry

Problem Statement

Find the smallest positive real number rr with the following property: For every choice of 20232023 unit vectors v1,v2,…,v2023∈R2v_1,v_2, \dots ,v_{2023} \in \mathbb{R}^2, a point pp can be found in the plane such that for each subset SS of {1,2,…,2023}\{1,2, \dots , 2023\}, the sum ∑i∈Svi\sum_{i \in S} v_i lies inside the disc {x∈R2:∣∣x−p∣∣≤r}\{x \in \mathbb{R}^2 : ||x-p|| \leq r\}.
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