2023 AIME I Problems/Problem 3

Problem

A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.

Solution

In this solution, let $\boldsymbol{n}$ -line points be the points where exactly $n$ lines intersect. We wish to find the number of $2$ -line points.

There are $\binom{40}{2}=780$ pairs of lines. Among them:

The $3$ -line points account for $3\cdot\binom32=9$ pairs of lines.

The $4$ -line points account for $4\cdot\binom42=24$ pairs of lines.

The $5$ -line points account for $5\cdot\binom52=50$ pairs of lines.

The $6$ -line points account for $6\cdot\binom62=90$ pairs of lines.

It follows that the $2$ -line points account for $780-9-24-50-90=\boxed{607}$ pairs of lines, where each pair intersect at a single point.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

~MRENTHUSIASM

2023 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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2023 AIME I Problems/Problem 3

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