1960 AHSME Problem 35

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}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.