{"status": "success", "data": {"description_md": "A cube-shaped container has vertices $A$, $B$, $C$, and $D$ where $\\overline{AB}$ and $\\overline{CD}$ are parallel edges of the cube, and $\\overline{AC}$ and $\\overline{BD}$ are diagonals of the faces of the cube. Vertex $A$ of the cube is set on a horizontal plane $\\mathcal P$ so that the plane of the rectangle $ABCD$ is perpendicular to $\\mathcal P$, vertex $B$ is $2$ meters above $\\mathcal P$, vertex $C$ is $8$ meters above $\\mathcal P$, and vertex $D$ is $10$ meters above $\\mathcal P$. The cube contains water whose surface is $7$ meters above $\\mathcal P$. The volume of the water is $\\tfrac mn$ cubic meters, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
$\\includegraphics[width=209, height=114, totalheight=114]{https://latex.artofproblemsolving.com/3/b/c/3bc5e7faf7f3526467427a84cf7c5b9df9d8bb18.png}$\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": "

A cube-shaped container has vertices A, B, C, and D where \\overline{AB} and \\overline{CD} are parallel edges of the cube, and \\overline{AC} and \\overline{BD} are diagonals of the faces of the cube. Vertex A of the cube is set on a horizontal plane \\mathcal P so that the plane of the rectangle ABCD is perpendicular to \\mathcal P, vertex B is 2 meters above \\mathcal P, vertex C is 8 meters above \\mathcal P, and vertex D is 10 meters above \\mathcal P. The cube contains water whose surface is 7 meters above \\mathcal P. The volume of the water is \\tfrac mn cubic meters, where m and n are relatively prime positive integers. Find m+n.

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 II Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/23_aime_II_p15", "prev": "/problem/23_aime_II_p13"}}