Problem:
A plane contains lines, no of which are parallel. Suppose that there are points where exactly lines intersect, points where exactly lines intersect, points where exactly lines intersect, points where exactly lines intersect, and no points where more than lines intersect. Find the number of points where exactly lines intersect.
Solution:
In this solution, let -line points be the points where exactly lines intersect. We wish to find the number of -line points.
There are pairs of lines. Among them:
It follows that the 2-line points account for pairs of lines, where each pair intersect at a single point.
The problems on this page are the property of the MAA's American Mathematics Competitions