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Mutually repulsive sets

Source: 2020 Israel Olympic Revenge P2

June 11, 2022
combinatoricsolympic revengeSets

Problem Statement

Let A,B⊂ZA, B\subset \mathbb{Z} be two sets of integers. We say that A,BA,B are mutually repulsive if there exist positive integers m,nm,n and two sequences of integers α1,α2,…,αn\alpha_1, \alpha_2, \dots, \alpha_n and β1,β2,…,βm\beta_1, \beta_2, \dots, \beta_m, for which there is a unique integer xx such that the number of its appearances in the sequence of sets A+α1,A+α2,…,A+αnA+\alpha_1, A+\alpha_2, \dots, A+\alpha_n is different than the number of its appearances in the sequence of sets B+β1,…,B+βmB+\beta_1, \dots, B+\beta_m.
For a given quadruple of positive integers (n1,d1,n2,d2)(n_1,d_1, n_2, d_2), determine whether the sets A={d1,2d1,…,n1d1}A=\{d_1, 2d_1, \dots, n_1d_1\} B={d2,2d2,…,n2d2}B=\{d_2, 2d_2, \dots, n_2d_2\} are mutually repulsive.
For a set X⊂ZX\subset \mathbb{Z} and c∈Zc\in \mathbb{Z}, we define X+c={x+c∣x∈X}X+c=\{x+c\mid x\in X\}.
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