{"status": "success", "data": {"description_md": "Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\\le 1$, $0<y\\le 1$, and $\\left[\\log_2{\\left(\\frac 1x\\right)}\\right]$ and $\\left[\\log_5{\\left(\\frac 1y\\right)}\\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.\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\">S</span> be the set of ordered pairs <span class=\"katex--inline\">(x, y)</span> such that <span class=\"katex--inline\">0&lt;x\\le 1</span>, <span class=\"katex--inline\">0&lt;y\\le 1</span>, and <span class=\"katex--inline\">\\left[\\log_2{\\left(\\frac 1x\\right)}\\right]</span> and <span class=\"katex--inline\">\\left[\\log_5{\\left(\\frac 1y\\right)}\\right]</span> are both even. Given that the area of the graph of <span class=\"katex--inline\">S</span> is <span class=\"katex--inline\">m/n</span>, where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers, find <span class=\"katex--inline\">m+n</span>. The notation <span class=\"katex--inline\">[z]</span> denotes the greatest integer that is less than or equal to <span class=\"katex--inline\">z</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": "2004 AIME I Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/04_aime_I_p13", "prev": "/problem/04_aime_I_p11"}}