{"status": "success", "data": {"description_md": "Let $\\mathcal{B}$ be the set of rectangular boxes that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\\mathcal{B}$ inside it. Then $r^{2}$ can be expressed as $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\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\">\\mathcal{B}</span> be the set of rectangular boxes that have volume <span class=\"katex--inline\">23</span> and surface area <span class=\"katex--inline\">54</span>. Suppose <span class=\"katex--inline\">r</span> is the least possible radius of a sphere that can fit any element of <span class=\"katex--inline\">\\mathcal{B}</span> inside it. Then <span class=\"katex--inline\">r^{2}</span> can be expressed as <span class=\"katex--inline\">\\tfrac{p}{q}</span>, where <span class=\"katex--inline\">p</span> and <span class=\"katex--inline\">q</span> are relatively prime positive integers. Find <span class=\"katex--inline\">p+q</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": 6, "problem_name": "2024 AIME I Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/24_aime_I_p14"}}