{"status": "success", "data": {"description_md": "Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\\frac{1}{2^m}$. What is the probability that Flora will eventually land at $10$?\n\n$\\textbf{(A) } \\frac{5}{512} \\qquad \\textbf{(B) } \\frac{45}{1024} \\qquad \\textbf{(C) } \\frac{127}{1024} \\qquad \\textbf{(D) } \\frac{511}{1024} \\qquad \\textbf{(E) } \\frac{1}{2}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Flora the frog starts at  <span class=\"katex--inline\">0</span>  on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance  <span class=\"katex--inline\">m</span>  with probability  <span class=\"katex--inline\">\\frac{1}{2^m}</span> . What is the probability that Flora will eventually land at  <span class=\"katex--inline\">10</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } \\frac{5}{512} \\qquad \\textbf{(B) } \\frac{45}{1024} \\qquad \\textbf{(C) } \\frac{127}{1024} \\qquad \\textbf{(D) } \\frac{511}{1024} \\qquad \\textbf{(E) } \\frac{1}{2}</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2023 AMC 12A Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12A_p18", "prev": "/problem/23_amc12A_p16"}}