IMO Shortlist 2016 problem A3


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Dodao/la: arhiva
3. listopada 2019.
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Find all positive integers n such that the following statement holds: Suppose real numbers a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n satisfy |a_k|+|b_k|=1 for all k=1,\dots,n. Then there exists \varepsilon_1, \varepsilon_2, \dots, \varepsilon_n, each of which is either -1 or 1, such that \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1.

Izvor: https://www.imo-official.org/problems/IMO2016SL.pdf