{"status": "success", "data": {"description_md": "Let $$z=\\frac{1+i}{\\sqrt{2}}.$$What is $$\\left(z^{1^2}+z^{2^2}+z^{3^2}+\\ldots+z^{{12}^2}\\right) \\cdot \\left(\\frac{1}{z^{1^2}}+\\frac{1}{z^{2^2}}+\\frac{1}{z^{3^2}}+\\ldots+\\frac{1}{z^{{12}^2}}\\right)?$$\n\n$\\textbf{(A) } 18 \\qquad \\textbf{(B) } 72-36\\sqrt2 \\qquad \\textbf{(C) } 36 \\qquad \\textbf{(D) } 72 \\qquad \\textbf{(E) } 72+36\\sqrt2$\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--display\">z=\\frac{1+i}{\\sqrt{2}}.</span> What is  <span class=\"katex--display\">\\left(z^{1^2}+z^{2^2}+z^{3^2}+\\ldots+z^{{12}^2}\\right) \\cdot \\left(\\frac{1}{z^{1^2}}+\\frac{1}{z^{2^2}}+\\frac{1}{z^{3^2}}+\\ldots+\\frac{1}{z^{{12}^2}}\\right)?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 18 \\qquad \\textbf{(B) } 72-36\\sqrt2 \\qquad \\textbf{(C) } 36 \\qquad \\textbf{(D) } 72 \\qquad \\textbf{(E) } 72+36\\sqrt2</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": "2019 AMC 12A Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12A_p22", "prev": "/problem/19_amc12A_p20"}}