CN-TST 2022 Problem 14

Two positive real numbers $\alpha, \beta$ satisfies that for any positive integers $k_1,k_2$ , it holds that $\lfloor k_1 \alpha \rfloor \neq \lfloor k_2 \beta \rfloor$ , where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$ . Prove that there exist positive integers $m_1,m_2$ such that $\frac{m_1}{\alpha}+\frac{m_2}{\beta}=1$ .