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m divides linear sum of sequences

Source: 2021 China Mathematical Olympiad P2

November 24, 2020
number theoryChina

Problem Statement

Let m>1m>1 be an integer. Find the smallest positive integer nn, such that for any integers a1,a2,,an;b1,b2,,bna_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n there exists integers x1,x2,,xnx_1,x_2,\ldots ,x_n satisfying the following two conditions:
i) There exists i{1,2,,n}i\in \{1,2,\ldots ,n\} such that xix_i and mm are coprime
ii) i=1naixii=1nbixi0(modm)\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod m