Difference between revisions of "2023 AIME I Problems/Problem 3"
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| − | A plane contains 40 lines, no 2 of which are parallel. Suppose that there are 3 points where exactly 3 lines | + | ==Problem== |
| − | 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 | + | 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. |
| − | exactly 2 lines intersect. | ||
==Solution== | ==Solution== | ||
| − | + | 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. | |
| − | exactly | + | |
| − | < | + | There are <math>\binom{40}{2}=780</math> pairs of lines. Among them: |
| − | + | ||
| − | + | * The <math>3</math>-line points account for <math>3\cdot\binom32=9</math> pairs of lines. | |
| − | - 5 \cdot \ | + | |
| − | + | * The <math>4</math>-line points account for <math>4\cdot\binom42=24</math> pairs of lines. | |
| − | + | ||
| − | </ | + | * The <math>5</math>-line points account for <math>5\cdot\binom52=50</math> pairs of lines. |
| + | |||
| + | * The <math>6</math>-line points account for <math>6\cdot\binom62=90</math> pairs of lines. | ||
| + | |||
| + | 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. | ||
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
| + | |||
| + | ~MRENTHUSIASM | ||
| + | |||
| + | ==Video Solution by TheBeautyofMath== | ||
| + | https://youtu.be/3fC11X0LwV8 | ||
| + | |||
| + | ~IceMatrix | ||
| + | |||
| + | ==See also== | ||
| + | {{AIME box|year=2023|num-b=2|num-a=4|n=I}} | ||
| + | {{MAA Notice}} | ||
| + | [[Category:Introductory Combinatorics Problems]] | ||
Latest revision as of 01:09, 10 December 2023
Problem
A plane contains 
lines, no 
of which are parallel. Suppose that there are 
points where exactly lines intersect, 
points where exactly lines intersect, 
points where exactly lines intersect, 
points where exactly lines intersect, and no points where more than lines intersect. Find the number of points where exactly lines intersect.
Solution
In this solution, let 
-line points be the points where exactly 
lines intersect. We wish to find the number of -line points.
There are 
pairs of lines. Among them:
- The
-line points account for 
pairs of lines.
pairs of lines.- The
-line points account for 
pairs of lines.
pairs of lines.- The
-line points account for 
pairs of lines.
pairs of lines.- The
-line points account for 
pairs of lines.
pairs of lines.It follows that the -line points account for 
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
~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. 
