A plane is divided into the maximum number of regions by 20 lines, none of which are parallel. How many regions does this divide the plane into at most? Among these, how many regions have a finite area?
We will determine how much the drawing of another line can increase the number of plane sections. The first line divides the plane into two parts. A second line, if it intersects the first, divides each of the two plane sections into one more, and thus two lines divide the plane into four parts. A third line, if it intersects the first two at different points, will further divide three plane sections, and thus multiplies the number of plane sections by 3. A fourth line can further divide as many plane sections as the points of intersection with the previous lines divide this line. The fourth line can therefore multiply the number of plane sections by up to 4, provided it intersects each of the previous lines but does not pass through any of their intersection points.

In general, if there are $n-1$ lines in the plane and we draw an $n$-th line, the previous lines will divide this line into segments (finite segments and two infinite rays extending from the extreme intersection points). Each segment of the line divides one plane section into two. The $n$-th line can have a maximum of $n-1$ intersection points with the previous lines (if it does not pass through any of their intersection points), and these points divide the line into $n$ segments. If we draw 20 lines one after another, the first two divide the plane into two parts, and the subsequent ones multiply the number of plane sections by $2, 3, 4, \ldots, 20$ respectively, provided there are no parallel lines and no three lines intersect at the same point. Thus, 20 lines

$$
2+2+3+4+\ldots+20=2+(2+20)+(3+19)+(4+18)+\ldots+(10+12)+11=2+9 \cdot 22+11=211
$$

divide the plane into 211 parts.

When drawing the lines one after another, infinite plane sections can only arise from infinite plane sections. When an infinite plane section is divided, both resulting parts will be infinite only if the segment of the dividing line that extends to infinity divides it. Since there are no parallel lines by the condition, each line doubles the number of infinite plane sections. Since the first line also divides the plane into 2 infinite parts, the number of infinite plane sections is twice the number of lines. Specifically, among the 211 plane sections created by 20 lines, 40 will be infinite.

Note: 1. We can similarly determine the maximum number of plane sections for any number of lines that can divide the plane. To count the plane sections, we needed to know how many segments the intersection points divide the line into. This reasoning can be extended to determine how many parts a given number of planes can divide space into.

2. The infinite plane sections can also be counted as follows. Surround all the intersection points of the lines with, for example, a sufficiently large circle. This circle will pass through the infinite plane sections, with one arc falling into each; since there are no parallel lines, no two arcs can fall into the same plane section, so there are as many infinite plane sections as the lines divide the circle. For $n$ lines, these will intersect the circle at $2n$ points and divide it into as many arcs, so for $n$ lines: $2n$, specifically for 20 lines: 40 infinite plane sections.

Remark: Many contestants confused the concept of "line" in the problem statement with the concept of "segment." Naturally, they could not find a solution.