{"status": "success", "data": {"description_md": "Let $P(z) = z^8 + \\left(4\\sqrt{3} + 6\\right)z^4 - \\left(4\\sqrt{3} + 7\\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?\n\n$\\textbf{(A)}\\ 4\\sqrt{3} + 4 \\qquad \\textbf{(B)}\\ 8\\sqrt{2} \\qquad \\textbf{(C)}\\  3\\sqrt{2} + 3\\sqrt{6} \\qquad \\textbf{(D)}\\  4\\sqrt{2} + 4\\sqrt{3} \\qquad \\textbf{(E)}\\  4\\sqrt{3} + 6$\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>Let  <span class=\"katex--inline\">P(z) = z^8 + \\left(4\\sqrt{3} + 6\\right)z^4 - \\left(4\\sqrt{3} + 7\\right)</span> . What is the minimum perimeter among all the  <span class=\"katex--inline\">8</span> -sided polygons in the complex plane whose vertices are precisely the zeros of  <span class=\"katex--inline\">P(z)</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 4\\sqrt{3} + 4 \\qquad \\textbf{(B)}\\ 8\\sqrt{2} \\qquad \\textbf{(C)}\\  3\\sqrt{2} + 3\\sqrt{6} \\qquad \\textbf{(D)}\\  4\\sqrt{2} + 4\\sqrt{3} \\qquad \\textbf{(E)}\\  4\\sqrt{3} + 6</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": "2011 AMC 12B Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/11_amc12B_p25", "prev": "/problem/11_amc12B_p23"}}