Find all integers $m$ such that $\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} $ How would you determine all integers $m$ such that the following is true? 
$$\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} .$$
Note that $\lfloor \cdot \rfloor$ means the greatest integer function. Also, $x$ must be a positive real number.

Consider the reciprocal equation 
$$ m = \dfrac{\lfloor 2x \rfloor \cdot \lfloor 5x \rfloor}
             {\lfloor 2x \rfloor + \lfloor 5x \rfloor}$$
Letting $x=n+\delta$ where $0 \le \delta < 1$, we get
$$ m = \dfrac{(2n + \lfloor 2\delta \rfloor)(5n + \lfloor 5\delta \rfloor)}
             {7n + \lfloor 2\delta \rfloor + \lfloor 5\delta \rfloor}$$
We list the possible values of m with respect to  $\delta$.
\begin{array}{|c|c|c|c|c|c|}
\delta \in & \lfloor 2\delta \rfloor & \lfloor 5\delta \rfloor
           & \lfloor 2\delta \rfloor + \lfloor 5\delta \rfloor
           & \lfloor 2\delta \rfloor \cdot \lfloor 5\delta \rfloor
           & m \\
\hline
[  0, 0.2) & 0 & 0 & 0 & 0 & \dfrac{10n}{7}\\
[0.2, 0.4) & 0 & 1 & 1 & 0 & \dfrac{10n^2+2n}{7n+1}\\
[0.4, 0.5) & 0 & 2 & 2 & 0 & \dfrac{10n^2+4n}{7n+2}\\
[0.5, 0.6) & 1 & 2 & 3 & 2 & \dfrac{10n^2+9n+2}{7n+3}\\
[0.6, 0.8) & 1 & 3 & 4 & 3 & \dfrac{10n^2+11n+3}{7n+4}\\
[0.8, 1.0) & 1 & 4 & 5 & 4 & \dfrac{10n^2+13n+4}{7n+5}\\
\hline
\end{array}
Now its just a matter of finding when those rational polynomials have integer solutions.
