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:
- We construct all lines in set
Since the lines in set
are parallel to each other, they have 
points of intersection.
- 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. - 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
| 1976 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 27 |
Followed by Problem 29 | |
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