1976 AHSME Problems/Problem 28: Difference between revisions
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== Solution == | == Solution == | ||
We partition <math>\{L_1,L_2,\dots,L_{100}\}</math> into three sets. Let | |||
<cmath>\begin{align*} | |||
X &= \{L_n\mid n\equiv0\pmod{4}\}, \\ | |||
Y &= \{L_n\mid n\equiv1\pmod{4}\}, \\ | |||
Z &= \{L_n\mid n\equiv2,3\pmod{4}\}, \\ | |||
\end{align*}</cmath> | |||
from which <math>|X|=|Y|=25</math> and <math>|Z|=50.</math> | |||
Any two distinct lines can have at most one point of intersection. We construct the sets one by one: | |||
<ol style="margin-left: 1.5em;"> | |||
<li>We construct all lines in set <math>X.</math> <p> | |||
Since all lines in set <math>X</math> are parallel to each other, they have <math>0</math> points of intersection. | |||
</li> | |||
<li>We construct all lines in set <math>Y.</math> <p> | |||
The lines in set <math>Y</math> have <math>1</math> point of intersection, namely <math>A.</math> <p> | |||
Moreover, each line in set <math>Y</math> can have <math>1</math> point of intersection with each line in set <math>X.</math> <p> | |||
At this point, we have <math>1+625=626</math> additional points of intersection. | |||
</li> | |||
<li>We construct all lines in set <math>Z.</math> <p> | |||
</li> | |||
</ol> | |||
== See also == | == See also == | ||
Revision as of 18:17, 8 September 2021
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 have at most one point of intersection. We construct the sets one by one:
- We construct all lines in set
Since all 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, namely 
Moreover, each line in set
can have point of intersection with each line in set At this point, we have
additional points of intersection.
- We construct all lines in set
are parallel to each other, they have 
points of intersection.
have 
point of intersection, namely 
additional points of intersection.
See also
| 1976 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 27 |
Followed by Problem 29 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
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