1976 AHSME Problems/Problem 28

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

To maximize the number of points of intersection, note that each point of intersection is 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 partition $L_1,L_2,\dots,L_{100}$ into three sets. Let $\begin{align*} A &= \{L_n\mid n\equiv0\pmod{4}\}, \\ B &= \{L_n\mid n\equiv1\pmod{4}\}, \\ C &= \{L_n\mid n\equiv2\text{ or }3\pmod{4}\}, \end{align*}$ from which $|A|=|B|=25$ and $|C|=50.$

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
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1976 AHSME Problems/Problem 28

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