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"
Line 1: Line 1:
−
Unofficial problem statement:
+
A plane contains 40 lines, no 2 of which are parallel. Suppose that there are 3 points where exactly 3 lines
−
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?
+
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==
 +
 
 +
The number of points where
 +
exactly 2 lines intersect is
 +
<cmath>
 +
\begin{align*}
 +
& \binom{40}{2} - 3 \cdot \binom{3}{2} - 4 \cdot \binom{4}{2}
 +
- 5 \cdot \binom{5}{2} - 6 \cdot \binom{6}{2}  \\
 +
& = \boxed{\textbf{(607) }} .
 +
\end{align*}
 +
</cmath>
 +
 
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Revision as of 13:37, 8 February 2023

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

The number of points where exactly 2 lines intersect is \begin{align*} & \binom{40}{2} - 3 \cdot \binom{3}{2} - 4 \cdot \binom{4}{2} - 5 \cdot \binom{5}{2} - 6 \cdot \binom{6}{2} \\ & = \boxed{\textbf{(607) }} . \end{align*}

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_AIME_I_Problems/Problem_3&oldid=188971"