{"status": "success", "data": {"description_md": "Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB=12;$ face $ABFG$ is a trapezoid with $\\overline{AB}$ parallel to $\\overline{GF},$ $BF=AG=8,$ and $GF=6;$ and face $CDE$ has $CE=DE=14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is 12. Given that $EG^2=p-q\\sqrt{r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$\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>Polyhedron <span class=\"katex--inline\">ABCDEFG</span> has six faces. Face <span class=\"katex--inline\">ABCD</span> is a square with <span class=\"katex--inline\">AB=12;</span> face <span class=\"katex--inline\">ABFG</span> is a trapezoid with <span class=\"katex--inline\">\\overline{AB}</span> parallel to <span class=\"katex--inline\">\\overline{GF},</span> <span class=\"katex--inline\">BF=AG=8,</span> and <span class=\"katex--inline\">GF=6;</span> and face <span class=\"katex--inline\">CDE</span> has <span class=\"katex--inline\">CE=DE=14.</span> The other three faces are <span class=\"katex--inline\">ADEG, BCEF,</span> and <span class=\"katex--inline\">EFG.</span> The distance from <span class=\"katex--inline\">E</span> to face <span class=\"katex--inline\">ABCD</span> is 12. Given that <span class=\"katex--inline\">EG^2=p-q\\sqrt{r},</span> where <span class=\"katex--inline\">p, q,</span> and <span class=\"katex--inline\">r</span> are positive integers and <span class=\"katex--inline\">r</span> is not divisible by the square of any prime, find <span class=\"katex--inline\">p+q+r.</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": "2002 AIME I Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/02_aime_I_p14"}}