Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Solution 1)
(Undo the unfinished solution.)
 
(2 intermediate revisions by 2 users not shown)
Line 3: Line 3:
 
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.
 
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.
  
−
==Solution 1==
+
==Solution==
−
 
 
−
We can see that for n lines, there should be <math>\frac{n*(n-1)}{2}</math> points where two lines intersect assuming that there are only points where two lines meet. Thus there should be <math>\frac{40*39}{2}=780</math> points where two lines intersect
 
−
 
 
−
==Solution 2==
 
  
 
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.
 
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.

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 (Problems • Answer Key • Resources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_AIME_I_Problems/Problem_3&oldid=206762"