In the plane, there are $m$ parallel lines and $n$ additional lines, none of which are parallel to each other or to the first set. Furthermore, we know that no point in the plane is intersected by more than two of these $m+n$ lines. How many regions do these lines divide the plane into?
Solution: Let's first deal with the non-parallel lines. Before drawing the first line, we already have one domain, i.e., the entire plane. The first line divides the plane into 2 parts, thus establishing one new domain. The second line intersects the first and passes through 2 domains, dividing each into two parts, thus establishing two new domains. The third line intersects the first two at two different, finite points, so it passes through 3 domains, each of which it divides into two parts, thus creating three new domains, and so on. Finally, the $n$-th line intersects the previous $n-1$ lines (at $n-1$ different, finite points, according to our conditions), thus dividing $n$ domains into two parts, creating $n$ new domains. Therefore, the number of domains after drawing the $n$-th line is:

$$
1+1+2+\ldots+(n-1)+n=1+\frac{n(n+1)}{2}
$$

Each of the $m$ parallel lines - according to our conditions - intersects the $n$ lines at $n$ different finite points, thus passing through $n+1$ domains, dividing each into two parts, and thus establishing $n+1$ new domains. Therefore, the $m$ parallel lines, in total, establish $m(n+1)$ new domains.

Therefore, the $m+n$ lines that meet our conditions divide the plane into:

$$
1+\frac{n(n+1)}{2}+m(n+1)
$$

domains.