Question 109
Let $[r]$ be the largest integer not exceeding real number $r$. For real positive numbers $x$, the set $A(x)$ is defined by
$$
A(x) = \{[nx] \mid n: \text{positive integer}\}.
$$
Find all irrational numbers $\alpha \gt 1$ satisfying the following condition.
Condition: If a positive real number $\beta$ satisfies $A(\alpha) \supset A(\beta)$, then $\frac{\beta}{\alpha}$ is an integer.
$$
A(x) = \{[nx] \mid n: \text{positive integer}\}.
$$
Find all irrational numbers $\alpha \gt 1$ satisfying the following condition.
Condition: If a positive real number $\beta$ satisfies $A(\alpha) \supset A(\beta)$, then $\frac{\beta}{\alpha}$ is an integer.