2004 AMC 12A Problem 23

A polynomial

P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0

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

\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.

Which of the following quantities can be a nonzero number?

\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}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.