{"status": "success", "data": {"description_md": "Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\\sqrt{21}$ and $\\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
$\\includegraphics[width=94, height=69, totalheight=69]{https://latex.artofproblemsolving.com/b/c/c/bcc8a67600e8a557236d37e2d01ae1737ef052b5.png}$\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": "

Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths \\sqrt{21} and \\sqrt{31}. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is \\frac mn, where m and n are relatively prime positive integers. Find m+n. A parallelepiped is a solid with six parallelogram faces such as the one shown below.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "2023 AIME I Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/23_aime_I_p14", "prev": "/problem/23_aime_I_p12"}}