2023 AIME I Problems/Problem 3
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 1
We can see that for n lines, there should be
Solution 2
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.
- The
-line points account for
pairs of lines.
- The
-line points account for
pairs of lines.
- The
-line points account for
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 | ||
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