{"status": "success", "data": {"description_md": "Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\\tfrac{\\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?\n\n$\\textbf{(A)}~20\\qquad\\textbf{(B)}~21\\qquad\\textbf{(C)}~22\\qquad\\textbf{(D)}~23\\qquad\\textbf{(E)}~24$\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>Consider the set of complex numbers  <span class=\"katex--inline\">z</span>  satisfying  <span class=\"katex--inline\">|1+z+z^{2}|=4</span> . The maximum value of the imaginary part of  <span class=\"katex--inline\">z</span>  can be written in the form  <span class=\"katex--inline\">\\tfrac{\\sqrt{m}}{n}</span> , where  <span class=\"katex--inline\">m</span>  and  <span class=\"katex--inline\">n</span>  are relatively prime positive integers. What is  <span class=\"katex--inline\">m+n</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}~20\\qquad\\textbf{(B)}~21\\qquad\\textbf{(C)}~22\\qquad\\textbf{(D)}~23\\qquad\\textbf{(E)}~24</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": 3, "problem_name": "2023 AMC 12A Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12A_p17", "prev": "/problem/23_amc12A_p15"}}