Difference between revisions of "2023 AIME I Problems/Problem 3"
MRENTHUSIASM (talk | contribs) (Added in more details.) |
MRENTHUSIASM (talk | contribs) (Undo the unfinished solution.) |
||
| (11 intermediate revisions by 5 users not shown) | |||
| Line 17: | Line 17: | ||
* The <math>6</math>-line points account for <math>6\cdot\binom62=90</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=607</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 | ~MRENTHUSIASM | ||
| + | |||
| + | ==Video Solution by TheBeautyofMath== | ||
| + | https://youtu.be/3fC11X0LwV8 | ||
| + | |||
| + | ~IceMatrix | ||
==See also== | ==See also== | ||
{{AIME box|year=2023|num-b=2|num-a=4|n=I}} | {{AIME box|year=2023|num-b=2|num-a=4|n=I}} | ||
| + | {{MAA Notice}} | ||
| + | [[Category:Introductory Combinatorics Problems]] | ||
Latest revision as of 02:09, 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. 
