That's actually really cool!

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?

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.

\frac{1}{\alpha}+\frac{1}{\beta}=1

\alpha

\beta

can only be algebraic irrational numbers

\frac{\beta}{\alpha}+\frac{\alpha}{\beta}=2

No such

\alpha

\beta

exists

\frac{\alpha}{\beta}=3