{"status": "success", "data": {"description_md": "A [[polynomial]]\n\n$$P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0$$<br>has [[real]] [[coefficient]]s with $c_{2004}\\not = 0$ and $2004$ distinct complex [[zero]]es $z_k = a_k + b_ki$, $1\\leq k\\leq 2004$ with $a_k$ and $b_k$ real, $a_1 = b_1 = 0$, and\n\n$$\\sum_{k = 1}^{2004}{a_k} = \\sum_{k = 1}^{2004}{b_k}.$$<br>Which of the following quantities can be a nonzero number?\n\n$\\text {(A) } c_0 \\qquad \\text {(B) } c_{2003} \\qquad \\text {(C) } b_2b_3...b_{2004} \\qquad \\text {(D) } \\sum_{k = 1}^{2004}{a_k} \\qquad \\text {(E) }\\sum_{k = 1}^{2004}{c_k}$\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>A [[polynomial]]</p>&#10;<p> <span class=\"katex--display\">P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0</span> <br/>has [[real]] [[coefficient]]s with  <span class=\"katex--inline\">c_{2004}\\not = 0</span>  and  <span class=\"katex--inline\">2004</span>  distinct complex [[zero]]es  <span class=\"katex--inline\">z_k = a_k + b_ki</span> ,  <span class=\"katex--inline\">1\\leq k\\leq 2004</span>  with  <span class=\"katex--inline\">a_k</span>  and  <span class=\"katex--inline\">b_k</span>  real,  <span class=\"katex--inline\">a_1 = b_1 = 0</span> , and</p>&#10;<p> <span class=\"katex--display\">\\sum_{k = 1}^{2004}{a_k} = \\sum_{k = 1}^{2004}{b_k}.</span> <br/>Which of the following quantities can be a nonzero number?</p>&#10;<p> <span class=\"katex--inline\">\\text {(A) } c_0 \\qquad \\text {(B) } c_{2003} \\qquad \\text {(C) } b_2b_3...b_{2004} \\qquad \\text {(D) } \\sum_{k = 1}^{2004}{a_k} \\qquad \\text {(E) }\\sum_{k = 1}^{2004}{c_k}</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": "2004 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12A_p24", "prev": "/problem/04_amc12A_p22"}}