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Difference between revisions of "1976 AHSME Problems/Problem 28"

(Created page with "== Problem == Lines <math>L_1,L_2,\dots,L_{100}</math> are distinct. All lines <math>L_{4n}, n</math> a positive integer, are parallel to each other. All lines <math>L_{4n-3}...")
 
 
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== Problem ==
 
== Problem ==
 
Lines <math>L_1,L_2,\dots,L_{100}</math> are distinct. All lines <math>L_{4n}, n</math> a positive integer, are parallel to each other.  
 
Lines <math>L_1,L_2,\dots,L_{100}</math> are distinct. All lines <math>L_{4n}, n</math> a positive integer, are parallel to each other.  
−
All lines <math>L_{4n-3}</math>, <math>n</math> a positive integer, pass through a given point <math>A</math>.
+
All lines <math>L_{4n-3}, n</math> a positive integer, pass through a given point <math>A.</math> The maximum number of points of intersection of pairs of lines from the complete set <math>\{L_1,L_2,\dots,L_{100}\}</math> is
−
The maximum number of points of intersection of pairs of lines from the complete set <math>\{L_1,L_2,\dots,L_{100}\}</math> is
 
  
 
<math>\textbf{(A) }4350\qquad
 
<math>\textbf{(A) }4350\qquad
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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 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:
 +
<ol style="margin-left: 1.5em;">
 +
  <li>We construct all lines in set <math>X.</math> <p>
 +
Since the 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 with each other, namely <math>A.</math> <p>
 +
Moreover, each line in set <math>Y</math> can intersect each line in set <math>X</math> once. So, there are <math>25^2=625</math> points of intersection. <p>
 +
<b>Now, we have <math>\boldsymbol{1+625=626}</math> additional points of intersection.</b> <p>
 +
</li>
 +
  <li>We construct all lines in set <math>Z.</math> <p>
 +
The lines in set <math>Z</math> can have <math>\binom{50}{2}=1225</math> points of intersection with each other. <p>
 +
Moreover, each line in set <math>Z</math> can intersect each line in sets <math>X</math> and <math>Y</math> once. So, there are <math>50^2=2500</math> points of intersection. <p>
 +
<b>Now, we have <math>\boldsymbol{1225+2500=3725}</math> additional points of intersection.</b> <p>
 +
</li>
 +
</ol>
 +
Together, the answer is <math>626+3725=\boxed{\textbf{(B) }4351}.</math>
 +
 +
~MRENTHUSIASM
  
 
== See also ==
 
== See also ==

Latest revision as of 01:47, 14 February 2024

Problem

Lines $L_1,L_2,\dots,L_{100}$ are distinct. All lines $L_{4n}, n$ a positive integer, are parallel to each other. All lines $L_{4n-3}, n$ a positive integer, pass through a given point $A.$ The maximum number of points of intersection of pairs of lines from the complete set $\{L_1,L_2,\dots,L_{100}\}$ is

$\textbf{(A) }4350\qquad \textbf{(B) }4351\qquad \textbf{(C) }4900\qquad \textbf{(D) }4901\qquad \textbf{(E) }9851$

Solution

We partition $\{L_1,L_2,\dots,L_{100}\}$ into three sets. Let \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*} from which $|X|=|Y|=25$ and $|Z|=50.$

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 $X.$

    Since the lines in set $X$ are parallel to each other, they have $0$ points of intersection.

  2. We construct all lines in set $Y.$

    The lines in set $Y$ have $1$ point of intersection with each other, namely $A.$

    Moreover, each line in set $Y$ can intersect each line in set $X$ once. So, there are $25^2=625$ points of intersection.

    Now, we have $\boldsymbol{1+625=626}$ additional points of intersection.

  3. We construct all lines in set $Z.$

    The lines in set $Z$ can have $\binom{50}{2}=1225$ points of intersection with each other.

    Moreover, each line in set $Z$ can intersect each line in sets $X$ and $Y$ once. So, there are $50^2=2500$ points of intersection.

    Now, we have $\boldsymbol{1225+2500=3725}$ additional points of intersection.

Together, the answer is $626+3725=\boxed{\textbf{(B) }4351}.$

~MRENTHUSIASM

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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