Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.5 AMC 10/12 Find the answer

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?
$\textbf{(A)}~\frac{1}{6}$ $\textbf{(B)}~\frac{1}{5}$ $\textbf{(C)}~\frac{\sqrt{3}}{8}$ $\textbf{(D)}~\frac{\arctan \frac{1}{2}}{\pi}$ $\textbf{(E)}~\frac{2\arcsin \frac{1}{4}}{\pi}$

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

2023AMC12BP20.png
Let Cyrus's starting position be $S$ . WLOG, let the place Cyrus lands at for his first jump be $O$ . From $O$ , Cyrus can reach all the points on $\odot O$ . The probability that Cyrus will land less than $1$ unit away from $S$ is $\frac{4 \alpha }{ 2 \pi}$ .
$\sin \alpha = \frac{ \frac12 }{2} = \frac14, \quad \alpha = \arcsin \frac14$
Therefore, the answer is
$\frac{4 \arcsin \frac14 }{ 2 \pi} = \boxed{\textbf{(E) } \frac{2 \arcsin \frac{1}{4}}{\pi}}$
~isabelchen

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.