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 $2000:$ $$2001 \cdot \frac{1}{2}\,x^{2000} - \binom{2001}{2}\frac{1}{4}\,x^{1999} + \cdots = 0.$$

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