Difference between revisions of "2023 AIME I Problems/Problem 3"

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Unofficial problem statement:
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==Problem==
In a plane there exists <math>40</math> lines, none of which are parallel. There are <math>3</math> points in which exactly <math>3</math> lines intersect, <math>4</math> points in which exactly <math>4</math> lines intersect, <math>5</math> points in which exactly <math>5</math> lines intersect, and <math>6</math> points in which exactly <math>6</math> lines intersect. There are no points in which more than <math>6</math> lines intersect. How many points are there in which exactly <math>2</math> lines intersect?
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A plane contains <math>40</math> lines, no <math>2</math> of which are parallel. Suppose that there are <math>3</math> points where exactly <math>3</math> lines intersect, <math>4</math> points where exactly <math>4</math> lines intersect, <math>5</math> points where exactly <math>5</math> lines intersect, <math>6</math> points where exactly <math>6</math> lines intersect, and no points where more than <math>6</math> lines intersect. Find the number of points where exactly <math>2</math> lines intersect.
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==Solution==
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In this solution, let <b><math>\boldsymbol{n}</math>-line points</b> be the points where exactly <math>n</math> lines intersect. We wish to find the number of <math>2</math>-line points.
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There are <math>\binom{40}{2}=780</math> pairs of lines. Among them:
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* The <math>3</math>-line points account for <math>3\cdot\binom32=9</math> pairs of lines.
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* The <math>4</math>-line points account for <math>4\cdot\binom42=24</math> pairs of lines.
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* The <math>5</math>-line points account for <math>5\cdot\binom52=50</math> pairs of lines.
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* The <math>6</math>-line points account for <math>6\cdot\binom62=90</math> pairs of lines.
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It follows that the <math>2</math>-line points account for <math>780-9-24-50-90=\boxed{607}</math> pairs of lines, where each pair intersect at a single point.
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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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~MRENTHUSIASM
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==Video Solution by TheBeautyofMath==
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https://youtu.be/3fC11X0LwV8
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~IceMatrix
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==See also==
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{{AIME box|year=2023|num-b=2|num-a=4|n=I}}
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{{MAA Notice}}
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[[Category:Introductory Combinatorics Problems]]

Latest revision as of 01:09, 10 December 2023

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

Video Solution by TheBeautyofMath

https://youtu.be/3fC11X0LwV8

~IceMatrix

See also

2023 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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