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Disjoint sequence implies "Beatty"

Source: 2022 China TST, Test 3 P2

April 30, 2022
floor functionnumber theoryalgebra

Problem Statement

Two positive real numbers α,β\alpha, \beta satisfies that for any positive integers k1,k2k_1,k_2, it holds that ⌊k1α⌋≠⌊k2β⌋\lfloor k_1 \alpha \rfloor \neq \lfloor k_2 \beta \rfloor, where ⌊x⌋\lfloor x \rfloor denotes the largest integer less than or equal to xx. Prove that there exist positive integers m1,m2m_1,m_2 such that m1α+m2β=1\frac{m_1}{\alpha}+\frac{m_2}{\beta}=1.
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