{"status": "success", "data": {"description_md": "Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelias position when she stops will be greater than $1$?\n\n$\\textbf{(A) }\\frac{1}{3} \\qquad \\textbf{(B) }\\frac{1}{2} \\qquad \\textbf{(C) }\\frac{2}{3} \\qquad \\textbf{(D) }\\frac{3}{4} \\qquad \\textbf{(E) }\\frac{5}{6}$", "description_html": "

Ant Amelia starts on the number line at 0 and crawls in the following manner. For n=1,2,3, Amelia chooses a time duration t_n and an increment x_n independently and uniformly at random from the interval (0,1). During the n th step of the process, Amelia moves x_n units in the positive direction, using up t_n minutes. If the total elapsed time has exceeded 1 minute during the n th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most 3 steps in all. What is the probability that Amelia’s position when she stops will be greater than 1 ?

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\\textbf{(A) }\\frac{1}{3} \\qquad \\textbf{(B) }\\frac{1}{2} \\qquad \\textbf{(C) }\\frac{2}{3} \\qquad \\textbf{(D) }\\frac{3}{4} \\qquad \\textbf{(E) }\\frac{5}{6}

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Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2022 AMC 10B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc10B_p24", "prev": "/problem/22_amc10B_p22"}}