{"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 coefficients with $c_{2004} \\neq 0$ and $2004$ distinct complex zeroes $z_k = a_k + b_ki$, $1\\leq k\\leq 2004$ with $a_k$ and $b_k$ real, $a_1 = b_1 = 0$, and\n\n$$\\sum_{k = 1}^{2004}{a_k} = \\sum_{k = 1}^{2004}{b_k}.$$Which of the following quantities can be a nonzero number?\n\n$\\text {(A) } c_0 \\qquad \\text {(B) } c_{2003} \\qquad \\text {(C) } b_2b_3\\cdots b_{2004} \\qquad \\text {(D) } \\sum\\limits_{k = 1}^{2004}{a_k} \\qquad \\text {(E) }\\sum\\limits_{k = 1}^{2004}{c_k}$\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 coefficients with <span class=\"katex--inline\">c_{2004} \\neq 0</span> and <span class=\"katex--inline\">2004</span> distinct complex zeroes <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>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\\cdots b_{2004} \\qquad \\text {(D) } \\sum\\limits_{k = 1}^{2004}{a_k} \\qquad \\text {(E) }\\sum\\limits_{k = 1}^{2004}{c_k}</span></p>&#10;<hr/>&#10;<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>&#10;", "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"}}