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Vectors in a tilted square

Source: 2016 IMO Shortlist A3

July 19, 2017
IMO ShortlistalgebraInequality

Problem Statement

Find all positive integers nn such that the following statement holds: Suppose real numbers a1a_1, a2a_2, …\dots, ana_n, b1b_1, b2b_2, …\dots, bnb_n satisfy ∣ak∣+∣bk∣=1|a_k|+|b_k|=1 for all k=1,…,nk=1,\dots,n. Then there exists ε1\varepsilon_1, ε2\varepsilon_2, …\dots, εn\varepsilon_n, each of which is either −1-1 or 11, such that ∣∑i=1nεiai∣+∣∑i=1nεibi∣≤1. \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1.
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