Answer: 500

Solution:

Expand $\left(\frac{1}{2} - x\right)^{2001}$ by the binomial theorem. Its leading term $(-x)^{2001} = -x^{2001}$ cancels the $x^{2001}$ in the equation, so what remains is a polynomial of degree $2000:$ $$\begin{aligned} &2001 \cdot \frac{1}{2}\,x^{2000} \\ &\quad {}- \binom{2001}{2}\frac{1}{4}\,x^{1999} + \cdots = 0. \end{aligned}$$

By Vieta's formulas, the sum of the $2000$ roots is $$\begin{aligned} \frac{\binom{2001}{2}/4}{2001/2} &= \frac{2001 \cdot 2000/8}{2001/2} \\ &= \frac{2000}{4} = 500. \end{aligned}$$