5. (7 points) Determine the number of points at which 10 lines intersect, given that only two of them are parallel and exactly three of these lines intersect at one point.
Answer: 42.

Solution. Let's number the lines so that lines 1, 2, and 3 intersect at one point (denote this point as $X$). List all possible pairs of lines (1 and 2, 1 and 3, 1 and $4, \ldots, 8$ and 9, 8 and 10, 9 and 10) and their points of intersection. There are a total of 45 pairs of lines (there are 9 pairs of the form 1 and $\ell$, 8 pairs of the form 2 and $\ell$, and so on; $9+8+7+6+5+4+3+2+1=45$). According to the problem, exactly two lines are parallel. Therefore, a total of 44 points of intersection will be listed. In this case, all points of intersection of the lines except $X$ will be listed exactly once, and point $X$ will appear three times: for the pairs of lines 1 and 2, 1 and 3, 2 and 3. Erase two extra letters $X$ from the list of points of intersection. There will be exactly 42 points left, and this time all points of intersection will be counted exactly once.

Criteria. Any correct solution: 7 points.

Correctly calculated the number of pairs of lines and provided the correct answer: 2 points.

Considered only special cases or provided the correct answer without explanation: 1 point.