VII OBM

If $A=0$, then we have $B\lfloor x\rfloor =B\lfloor y\rfloor$ which obviously has infinitely many solutions with $x\ne y$. So assume $A\ne 0$. Then we can write the equation as $\frac{B}{A}(\lfloor x\rfloor -\lfloor y\rfloor )=y-x$. If $x\ne y$, we can assume $x<y$. We cannot have $\lfloor x\rfloor =\lfloor y\rfloor$, since $y-x>0$, so $\lfloor x\rfloor <\lfloor y\rfloor$. Write $x=n-ϵ$, $y=n+d+\delta$, where $n$ is any integer, $d$ is a non-negative integer, $\delta \in [0,1)$ and $ϵ\in (0,1]$. So $\frac{B}{A}=-\frac{d+\nu }{d+1}$, where $d$ is a non-negative integer and $\nu =\delta +ϵ\in (0,2)$.

We note first that $-2<-\frac{d+\nu }{d+1}<0$. So if there are solutions with $x\ne y$ and $A\ne 0$, then $\frac{B}{A}$ must belong to the open interval $(-2,0)$.

Conversely, by taking $d=0$ and suitable $\nu$ it is clear that $\frac{B}{A}$ can take any value in $(-2,0)$. Thus a necessary and sufficient condition for solutions with $x\ne y$ is $A=0$ or $\frac{B}{A}\in (-2,0)$.