1976 AHSME Problems/Problem 28

Problem

Lines are distinct. All lines a positive integer, are parallel to each other. All lines a positive integer, pass through a given point The maximum number of points of intersection of pairs of lines from the complete set is

Solution

We partition into three sets. Let from which and

Any two distinct lines can intersect at most once. To maximize the number of points of intersection, note that each point must be passed by exactly two lines. If three or more lines pass through the same point, then we can create more points of intersection by translating the lines.

We construct the sets one by one:

1. We construct all lines in set

   Since the lines in set are parallel to each other, they have points of intersection.

2. We construct all lines in set

   The lines in set have point of intersection with each other, namely

   Moreover, each line in set can intersect each line in set once. So, there are points of intersection.

   Now, we have additional points of intersection.

3. We construct all lines in set

   The lines in set can have points of intersection with each other.

   Moreover, each line in set can intersect each line in sets and once. So, there are points of intersection.

   Now, we have additional points of intersection.

Together, the answer is

~MRENTHUSIASM

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.