{"status": "success", "data": {"description_md": "Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a,b,c,$ and $d$ are integers, $0\\le d\\le c\\le b\\le a\\le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1.$ What is the sum of all values $P(1)$ over all the polynomials with these properties?\n\n$\\textbf{(A)}\\ 84\\qquad\\textbf{(B)}\\ 92\\qquad\\textbf{(C)}\\ 100\\qquad\\textbf{(D)}\\ 108\\qquad\\textbf{(E)}\\ 120$\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 all polynomials of a complex variable,  <span class=\"katex--inline\">P(z)=4z^4+az^3+bz^2+cz+d</span> , where  <span class=\"katex--inline\">a,b,c,</span>  and  <span class=\"katex--inline\">d</span>  are integers,  <span class=\"katex--inline\">0\\le d\\le c\\le b\\le a\\le 4</span> , and the polynomial has a zero  <span class=\"katex--inline\">z_0</span>  with  <span class=\"katex--inline\">|z_0|=1.</span>  What is the sum of all values  <span class=\"katex--inline\">P(1)</span>  over all the polynomials with these properties?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 84\\qquad\\textbf{(B)}\\ 92\\qquad\\textbf{(C)}\\ 100\\qquad\\textbf{(D)}\\ 108\\qquad\\textbf{(E)}\\ 120</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": "2012 AMC 12B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12B_p24", "prev": "/problem/12_amc12B_p22"}}