Question 23
A polynomial $$
\begin{aligned} &P(x) = c_{2004} x^{2004} + c_{2003} x^{2003} \\ &\quad {}+ \cdots + c_1 x + c_0 \end{aligned}
$$ has real coefficients with $c_{2004} \ne 0$ and $2004$ distinct complex zeros $z_k = a_k + b_k i,$ $1 \le k \le 2004$ 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?
\begin{aligned} &P(x) = c_{2004} x^{2004} + c_{2003} x^{2003} \\ &\quad {}+ \cdots + c_1 x + c_0 \end{aligned}
$$ has real coefficients with $c_{2004} \ne 0$ and $2004$ distinct complex zeros $z_k = a_k + b_k i,$ $1 \le k \le 2004$ with $a_k$ and $b_k$ real, $a_1 = b_1 = 0,$ and $$
\sum_{k=1}^{2004} a_k = \sum_{k=1}^{2004} b_k.
$$ Which of the following quantities can be a nonzero number?