2019 AMC 10 A

It is possible to obtain , , , , , and points of intersection, as demonstrated in the following figures:

It is clear that the maximum number of possible intersections is , since each pair of lines can intersect at most once. We now prove that it is impossible to obtain two intersections.

We proceed by contradiction. Assume a configuration of four lines exists such that there exist only two intersection points. Let these intersection points be and . Consider two cases:

Case : No line passes through both and

Then, since an intersection is obtained by an intersection between at least two lines, two lines pass through each of and . Then, since there can be no additional intersections, no line that passes through can intersect a line that passes through , and so each line that passes through must be parallel to every line that passes through . Then the two lines passing through are parallel to each other by transitivity of parallelism, so they coincide, contradiction.

Case : There is a line passing through and

Then there must be a line passing through , and a line passing through . These lines must be parallel. The fourth line must pass through either or . Without loss of generality, suppose passes through . Then since and cannot coincide, they cannot be parallel.

Then and cannot be parallel either, so they intersect, contradiction.

All possibilities have been exhausted, and thus we can conclude that two intersections is impossible.Our answer is given by the sum .