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2017 CNMO Grade 11 P4

Source: 2017 China Northern MO, Grade 11, Problem 4

February 24, 2020
number theory

Problem Statement

Positive intenger n≥3n\geq3. a1,a2,⋯ ,ana_1,a_2,\cdots,a_n are nn positive intengers that are pairwise coprime, satisfying that there exists k1,k2,⋯ ,kn∈{−1,1},∑i=1nkiai=0k_1,k_2,\cdots,k_n\in\{-1,1\}, \sum_{i=1}^{n}k_ia_i=0. Are there positive intengers b1,b2,⋯ ,bnb_1,b_2,\cdots,b_n, for any k∈Z+k\in\mathbb{Z}_+, b1+ka1,b2+ka2,⋯ ,bn+kanb_1+ka_1,b_2+ka_2,\cdots,b_n+ka_n are pairwise coprime?
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