In the unit square $ABCD$, points $E$ and $F$ are on the side $AB$ such that $AE = \frac{1}{n}$ and $BF = \frac{1}{m}$, where $m$ and $n$ are positive integers. A line starting from point $B$ and forming a $30^{\circ}$ angle with $AB$ is tangent to the semicircle drawn inside the square with $EF$ as its diameter. Determine the values of $m$ and $n$. Are there any pairs of values for $m$ and $n$ such that the semicircle is also tangent to the diagonal $AC$?
We denote the center of the semicircle by $O$, its radius by $r$, and the point of tangency by $T$. We distinguish two cases. Case I: $A EA F$ (Figure 3). Similarly to the previous case, we obtain that $B E=r$,

$$
r=\frac{E F}{2}=\frac{1}{2}(A E+B F-A B)=\frac{1}{2}\left(\frac{1}{n}+\frac{1}{m}-1\right)
$$

and $B E=A B-A E=1-\frac{1}{n}$. Therefore, $1-\frac{1}{n}=\frac{1}{2}\left(\frac{1}{n}+\frac{1}{m}-1\right)$, which simplifies to $3 m n-3 m-n=0$. Rearranging this, we get $(3 m-1)(n-1)=1$, which implies $3 m-1= \pm 1$, and thus there is no solution in positive integers in this case.

Therefore, $m$ and $n$ can take two different values: either $m=6$ and $n=2$, or $m=n=4$. In neither case does the diagonal $A C$ touch the semicircle.