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Algebra Level 5

\[\lfloor \alpha \rfloor,\lfloor 2\alpha \rfloor, \lfloor 3\alpha \rfloor, \ldots \quad \text{ and } \quad \lfloor \beta \rfloor, \lfloor 2\beta \rfloor,\lfloor 3\beta \rfloor, \ldots\]

Let \(\alpha\) and \(\beta\) be positive irrational numbers such that the sequences above contain every positive integer exactly once. i.e. If you pick a positive integer, it will appear in only one of the above sequences. Which of the following statements are true?

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Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

\[\frac{\alpha}{\beta}=3\] \[\frac{\beta}{\alpha}+\frac{\alpha}{\beta}=2\] \[\frac{1}{\alpha}+\frac{1}{\beta}=1\] No such \(\alpha\), \(\beta\) exists \(\alpha\), \(\beta\) can only be algebraic irrational numbers

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