Difference between revisions of "2025 AIME I Problems/Problem 13"
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Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws <math>25</math> more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these <math>27</math> line segments divide the disk. | Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws <math>25</math> more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these <math>27</math> line segments divide the disk. | ||
| + | |||
| + | ==Solution 1== | ||
| + | First, we calculate the probability that two segments intersect each other. Let the quadrants be numbered 1 through 4 in the normal labeling of quadrants and let the two segments be <math>A</math> and <math>B.</math> | ||
| + | [b]Case 1:[/b] Segment <math>A</math> has endpoints in two opposite quadrants. This happens with probability <math>\frac{1}{3}.</math> WLOG let the two quadrants be <math>1</math> and <math>3.</math> We do cases in which quadrants segment <math>B</math> lies in. | ||
| + | [list] | ||
| + | [*] | ||
| + | [/list] | ||
| + | (Work in Progress) | ||
==See also== | ==See also== | ||
Revision as of 20:54, 13 February 2025
Problem
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws 
more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these 
line segments divide the disk.
Solution 1
First, we calculate the probability that two segments intersect each other. Let the quadrants be numbered 1 through 4 in the normal labeling of quadrants and let the two segments be 
and 
[b]Case 1:[/b] Segment has endpoints in two opposite quadrants. This happens with probability 
WLOG let the two quadrants be 
and 
We do cases in which quadrants segment 
lies in.
[list]
[*]
[/list]
(Work in Progress)
See also
| 2025 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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. 
