Difference between revisions of "2023 AMC 12B Problems/Problem 20"
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Revision as of 22:09, 15 November 2023
Problem
Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position?
Solution 1
Denote by 
the position after the 
th jump.
Thus, to fall into the region centered at 
and with radius 1, 
.
Therefore, the probability is
![\[ \frac{2 \cdot 2 \arcsin \frac{1}{4}}{2 \pi} = \boxed{\textbf{(E) } \frac{2 \arcsin \frac{1}{4}}{\pi}}. \]](http://latex.artofproblemsolving.com/6/e/d/6edb5c671433291b14a0e16f099947181f8a4717.png)
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 2
(Diagram in progress......) (Writing in progress......)
Solution 3(coord bash)
Let the orgin be the starting point (writing continueing) ~ddk001
See Also
| 2023 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 19 |
Followed by Problem 21 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
