{"status": "success", "data": {"description_md": "The five solutions to the equation$$(z-1)(z^2+2z+4)(z^2+4z+6)=0$$ may be written in the form $x_k+y_ki$ for $1\\le k\\le 5,$ where $x_k$ and $y_k$ are real. Let $\\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\\mathcal E$ can be written in the form $\\sqrt{\\frac mn}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? (Recall that the eccentricity of an ellipse $\\mathcal E$ is the ratio $\\frac ca$, where $2a$ is the length of the major axis of $\\mathcal E$ and $2c$ is the is the distance between its two foci.)\n\n$\\textbf{(A) }7 \\qquad \\textbf{(B) }9 \\qquad \\textbf{(C) }11 \\qquad \\textbf{(D) }13\\qquad \\textbf{(E) }15$\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": "

The five solutions to the equation (z-1)(z^2+2z+4)(z^2+4z+6)=0 may be written in the form x_k+y_ki for 1\\le k\\le 5, where x_k and y_k are real. Let \\mathcal E be the unique ellipse that passes through the points (x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4), and (x_5,y_5) . The eccentricity of \\mathcal E can be written in the form \\sqrt{\\frac mn} , where m and n are relatively prime positive integers. What is m+n ? (Recall that the eccentricity of an ellipse \\mathcal E is the ratio \\frac ca , where 2a is the length of the major axis of \\mathcal E and 2c is the is the distance between its two foci.)

\\textbf{(A) }7 \\qquad \\textbf{(B) }9 \\qquad \\textbf{(C) }11 \\qquad \\textbf{(D) }13\\qquad \\textbf{(E) }15

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": "2021 AMC 12A Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12A_p22", "prev": "/problem/21_amc12A_p20"}}