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:
Answer (607):
There are pairs of lines, and each pair has 1 intersection point. There are pairs of lines intersecting at points where 6 lines intersect, pairs of lines intersecting at points where 5 lines intersect, pairs of lines intersecting at points where 4 lines intersect, and pairs of lines intersecting at points where 3 lines intersect. This accounts for of the 780 pairs of lines. Therefore there are points where exactly 2 of the lines intersect.
To see that the claimed configuration of 40 lines is possible, draw 6 lines , and , where no 2 of the lines are parallel and no 3 of them intersect at the same point. On these lines select 18 points labeled , and , where, for each index, points with that index lie on the line with the same index, and where 3 of the selected points are collinear only if the 3 points lie on 1 of the 6 drawn lines. For all valid choices of , draw the lines , and . When each new point is selected, there are infinitely many points from which to choose and only finitely many points that would result in parallel lines or new intersection points where more than 2 lines intersect. Thus it is possible to arrange the selected points so that the only points where more than 2 lines intersect are the 18 labeled points and no 2 of the drawn lines are parallel. This accounts for lines. Each of the points lies on 6 of the drawn lines, each of the points lies on 5 of the drawn lines, each of the points lies on 4 of the drawn lines, and each of the points lies on 3 of the drawn lines, as required.
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions