From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. 
Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. 
It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. 
If $m$ and $t$ are integers, then $t$ may have the following number of values:
$\textbf{(A)}\ \text{zero}\qquad \textbf{(B)}\ \text{one}\qquad \textbf{(C)}\ \text{two}\qquad \textbf{(D)}\ \text{three}\qquad \textbf{(E)}\ \text{infinitely many}$
By definition of [mean proportional](https://artofproblemsolving.com/wiki/index.php/Geometric_mean), $t = \sqrt{mn}$.  Since $m+n=10$, $t = \sqrt{m(10-m)}$.
With trial and error, note that when $m=9$, $t=3$ and when $m=2$, $t=4$.  These values work since another [tangent line](https://artofproblemsolving.com/wiki/index.php/Tangent_line) can be drawn from $P$, and the angle between the [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent_line) and [secant](https://artofproblemsolving.com/wiki/index.php/Secant_line) can decrease to match the values of $m$ and $n$.  Thus, the answer is $\boxed{\textbf{(C)}}$.