{"status": "success", "data": {"description_md": "Given a real number $x,$ let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \\ldots, n_{70}$ such that $k=\\lfloor\\sqrt[3]{n_{1}}\\rfloor = \\lfloor\\sqrt[3]{n_{2}}\\rfloor = \\cdots = \\lfloor\\sqrt[3]{n_{70}}\\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \\leq i \\leq 70.$<br>Find the maximum value of $\\frac{n_{i}}{k}$ for $1\\leq i \\leq 70.$\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>Given a real number <span class=\"katex--inline\">x,</span> let <span class=\"katex--inline\">\\lfloor x \\rfloor</span> denote the greatest integer less than or equal to <span class=\"katex--inline\">x.</span> For a certain integer <span class=\"katex--inline\">k,</span> there are exactly <span class=\"katex--inline\">70</span> positive integers <span class=\"katex--inline\">n_{1}, n_{2}, \\ldots, n_{70}</span> such that <span class=\"katex--inline\">k=\\lfloor\\sqrt[3]{n_{1}}\\rfloor = \\lfloor\\sqrt[3]{n_{2}}\\rfloor = \\cdots = \\lfloor\\sqrt[3]{n_{70}}\\rfloor</span> and <span class=\"katex--inline\">k</span> divides <span class=\"katex--inline\">n_{i}</span> for all <span class=\"katex--inline\">i</span> such that <span class=\"katex--inline\">1 \\leq i \\leq 70.</span><br/>Find the maximum value of <span class=\"katex--inline\">\\frac{n_{i}}{k}</span> for <span class=\"katex--inline\">1\\leq i \\leq 70.</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": 4, "problem_name": "2007 AIME II Problem 7", "can_next": true, "can_prev": true, "nxt": "/problem/07_aime_II_p08", "prev": "/problem/07_aime_II_p06"}}