How to find the places where this interval encloses an integer Suppose I have an interval that looks like this:
$\left[\frac{k}{\lfloor m \rfloor }, \frac{kr}{\lfloor mr \rfloor}\right)$
$m$ and $r$ are positive real numbers, but are constants in this problem.
Here is the question:
Which integer values of $k$ make this interval include at least one integer?
Note that this is very similar to the problem I asked this morning here, but I actually copied it down from my sheet of paper wrong.  The problem that I am trying to solve here has an additional $r$ in the second term.
I have tried various ways to tackle this problem, but none have panned out.  The thing is I have no idea where to even start.  How does one solve problems like this?
 A: If the constants $m,r$ are actually specified, the set of qualifying $k$-values can be readily found, but with unknown constants $m,r$, it doesn't look like an easy problem.

However, for the existence of a qualifying $k$-value, there are some easy necessary conditions, as well as an easy sufficient condition . .  .

First, some necessary conditions:

We must have 
$$m \ge 1\tag{1}$$
$$mr \ge 1\tag{2}$$
to ensure that both endpoints of the interval are defined.

Also, the left endpoint of the interval must be less than the right endpoint, hence
$$\frac{k}{\lfloor{m}\rfloor} < \frac{kr}{\lfloor{mr}\rfloor}$$
Dividing both sides by $k$, yields the necessary condition
$$\lfloor{mr}\rfloor < \lfloor{m}\rfloor{r}\tag{3}$$
Next, a sufficient condition:

If the right endpoint is at least 1 more than the left endpoint, then the interval is guaranteed to contain an integer, hence it will suffice to have
$$\frac{kr}{\lfloor{mr}\rfloor} - \frac{k}{\lfloor m \rfloor} \ge 1$$
which yields the sufficient condition
$$k \ge 
\frac
{{\lfloor{m}\rfloor}{\lfloor{mr}\rfloor}}
{{\lfloor{m}\rfloor{r}}-{\lfloor{mr}\rfloor}}
\tag{4}
$$
Thus, for any choice of constants $m,r$ satisfying the necessary conditions, any value of $k$ greater or equal to
$$
\left
\lceil
{
\frac
{{\lfloor{m}\rfloor}{\lfloor{mr}\rfloor}}
{{\lfloor{m}\rfloor{r}}-{\lfloor{mr}\rfloor}}
}
\right
\rceil
$$
will be a qualifying $k$-value.

However, it's possible for a value of $k$ to be qualifying even if condition $(4)$ is not satisfied. For example, if
$$k=8,\;\;m=\frac{7}{2},\;\;r=\frac{11}{14}$$
then the integer $k=8$ yields the interval 
$$\left[\frac{8}{3},\frac{22}{7}\right)$$
which contains the integer $3$, hence for the constants $m,r$ specified above, $8$ is a qualifying $k$-value, even though condition $(4)$ is not satisfied.

Other than the necessary conditions $(1),(2),(3)$, and the sufficient condition $(4)$, I don't know how much more can be said without more information about the constants $m,r$.

For example, suppose $r$ is required to be a positive integer.

Let $s = \lfloor{m}\rfloor$. Then
\begin{align*}
&s = \lfloor{m}\rfloor\\[4pt]
\implies\; &s \le m\\[4pt]
\implies\; &sr \le mr\\[4pt]
\implies\; &\lfloor{sr}\rfloor \le \lfloor{mr}\rfloor\\[4pt]
\implies\; &sr \le \lfloor{mr}\rfloor\qquad\text{[since $s,r$ are integers]}\\[4pt]
\implies\; &\lfloor{m}\rfloor r \le \lfloor{mr}\rfloor\\[4pt]
\end{align*}
contrary to necessary condition $(3)$.

Hence, if $r$ is required to be a positive integer, the interval is empty, so there are no qualifying values of $k$.
