209. Indicate all such numbers $\alpha$ that the numbers $[\alpha],[2 \alpha]$, $[3 \alpha], \ldots,[N \alpha]$, where $N$ is a fixed natural number, are all distinct, and the numbers $\left[\frac{1}{\alpha}\right],\left[\frac{2}{\alpha}\right],\left[\frac{3}{\alpha}\right], \ldots,\left[\frac{N}{\alpha}\right]$ are also all distinct.

In problem 209, it is required to find such a number $\alpha$ so that the numbers $[\alpha],[2 \alpha],[3 \alpha], \ldots,[N \alpha]$ and $[\beta],[2 \beta],[3 \beta], \ldots,[N \beta]$, where $\beta=\frac{1}{\alpha}$, are all distinct. A more substantial problem is to determine such $\alpha$ and $\beta$ (where it is not necessarily $\beta=\frac{1}{\alpha}$) so that the infinite sequences $[\alpha],[2 \alpha],[3 \alpha], \ldots ;[\beta],[2 \beta],[3 \beta], \ldots$ consist of pairwise distinct numbers; for example, it can be proven that the listed sequences contain all natural numbers, and each number appears exactly once, if and only if $\alpha$ is irrational and $\frac{1}{\alpha}+\frac{1}{\beta}=1$.
209. Let $\alpha>0$. It is clear that the value $\alpha=1$ satisfies the condition of the problem; if, however, $\alpha>1$ and, accordingly, $\frac{1}{\alpha}<1$, then $\frac{N-1}{N} \leqslant \beta=\frac{1}{\alpha}<1$; if $\alpha<1$ and, accordingly, $\frac{1}{\alpha}>1$, then $\frac{N-1}{N} \leqslant \beta=\frac{1}{\alpha}>1$. Therefore, for $\alpha>0$ it must be $\frac{N-1}{N} \leqslant \alpha \leqslant \frac{N}{N-1}$; similarly, it is established that for $\alpha<0$ it must be $-\frac{N-1}{N} \geqslant \alpha \geqslant-\frac{N}{N-1}$.