We work in the complex plane and without loss of generality we assume the charges are placed at the locations
$\begin{align*}r e^{i 2 \pi j/5} \mbox{\ \ \ \ \ \ \ \ j = 0,1,2,3,4}\end{align*}$
The electric field at the center is then given by Coulomb's Law:
$\begin{align*}E_x + E_y i = \sum_{j = 0}^4 \frac{q}{4 \pi \epsilon_0} \frac{-e^{i 2 \pi j/5}}{r^2} = \frac{-q}{4 \pi \epsilon...$
We use the following formula for a geometric series:
$\begin{align*}\sum_{j = 0}^n x^j = \frac{1 - x^{j+1}}{1-x}\end{align*}$
where $n$ is a positive integer and $x$ is any complex number not equal to 1. Letting $x = e^{i 2 \pi/5}$ and $n= 4$ , we find that
$\begin{align*}\sum_{j = 0}^4 e^{i 2 \pi j/5} = \sum_{j = 0}^4 \left(e^{i 2 \pi/5}\right)^j = \frac{1 - \left(e^{i 2 \pi/5}\ri...$
Therefore,
$\begin{align*}E_x + E_y i = 0\end{align*}$
at the origin. Therefore, $\left|\mathbf{E}\right| = 0$ at the origin. Therefore, answer (A) is correct.

Solution to 2001 Problem 9