{"status": "success", "data": {"description_md": "Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. What is $m + n + p$?\n\n$\\textbf{(A)}\\ 602\\qquad \\textbf{(B)}\\ 652\\qquad \\textbf{(C)}\\ 702\\qquad \\textbf{(D)}\\ 752 \\qquad \\textbf{(E)}\\ 802$\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": "

Functions f and g are quadratic, g(x) = - f(100 - x) , and the graph of g contains the vertex of the graph of f . The four x -intercepts on the two graphs have x -coordinates x_1 , x_2 , x_3 , and x_4 , in increasing order, and x_3 - x_2 = 150 . Then x_4 - x_1 = m + n\\sqrt p , where m , n , and p are positive integers, and p is not divisible by the square of any prime. What is m + n + p ?

\\textbf{(A)}\\ 602\\qquad \\textbf{(B)}\\ 652\\qquad \\textbf{(C)}\\ 702\\qquad \\textbf{(D)}\\ 752 \\qquad \\textbf{(E)}\\ 802

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": 5, "problem_name": "2009 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/09_amc12A_p24", "prev": "/problem/09_amc12A_p22"}}