{"status": "success", "data": {"description_md": "Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full 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>Let <span class=\"katex--inline\">A</span>, <span class=\"katex--inline\">B</span>, <span class=\"katex--inline\">C</span>, and <span class=\"katex--inline\">D</span> be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex <span class=\"katex--inline\">A</span>, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let <span class=\"katex--inline\">p = n/729</span> be the probability that the bug is at vertex <span class=\"katex--inline\">A</span> when it has crawled exactly 7 meters. Find the value of <span class=\"katex--inline\">n</span>.</p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. 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": 5, "problem_name": "1985 AIME Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/85_aime_p13", "prev": "/problem/85_aime_p11"}}