{"status": "success", "data": {"description_md": "Let $G$ be the set of polynomials of the form\n\n$$P(z)=z^n+c_{n-1}z^{n-1}+\\cdots+c_2z^2+c_1z+50,$$<br>where $c_1,c_2,\\cdots, c_{n-1}$ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?\n\n$\\textbf{(A)}\\ 288\\qquad\\textbf{(B)}\\ 528\\qquad\\textbf{(C)}\\ 576\\qquad\\textbf{(D)}\\ 992\\qquad\\textbf{(E)}\\ 1056$\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\">G</span>  be the set of polynomials of the form</p>&#10;<p> <span class=\"katex--display\">P(z)=z^n+c_{n-1}z^{n-1}+\\cdots+c_2z^2+c_1z+50,</span> <br/>where  <span class=\"katex--inline\">c_1,c_2,\\cdots, c_{n-1}</span>  are integers and  <span class=\"katex--inline\">P(z)</span>  has distinct roots of the form  <span class=\"katex--inline\">a+ib</span>  with  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  integers. How many polynomials are in  <span class=\"katex--inline\">G</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 288\\qquad\\textbf{(B)}\\ 528\\qquad\\textbf{(C)}\\ 576\\qquad\\textbf{(D)}\\ 992\\qquad\\textbf{(E)}\\ 1056</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": "2013 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/13_amc12B_p24"}}