{"status": "success", "data": {"description_md": "A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\\frac {a\\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. What is $a + b + c$?\n\n$\\textbf{(A)}\\ 10\\qquad \\textbf{(B)}\\ 11\\qquad \\textbf{(C)}\\ 12\\qquad \\textbf{(D)}\\ 13\\qquad \\textbf{(E)}\\ 14$\n___\nFull 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>A regular octahedron has side length  <span class=\"katex--inline\">1</span> . A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area  <span class=\"katex--inline\">\\frac {a\\sqrt {b}}{c}</span> , where  <span class=\"katex--inline\">a</span> ,  <span class=\"katex--inline\">b</span> , and  <span class=\"katex--inline\">c</span>  are positive integers,  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">c</span>  are relatively prime, and  <span class=\"katex--inline\">b</span>  is not divisible by the square of any prime. What is  <span class=\"katex--inline\">a + b + c</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 10\\qquad \\textbf{(B)}\\ 11\\qquad \\textbf{(C)}\\ 12\\qquad \\textbf{(D)}\\ 13\\qquad \\textbf{(E)}\\ 14</span> </p>&#10;<hr><p>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": 5, "problem_name": "2009 AMC 12A Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/09_amc12A_p23", "prev": "/problem/09_amc12A_p21"}}