Problem 2. Calculate the value of the expression $\frac{x^{4}-2005 x^{3}+2005 x^{2}-x}{x-1}$ for $x=2005$.
Solution. Let's simplify the expression $A(x)=\frac{x^{4}-2005 x^{3}+2005 x^{2}-x}{x-1}$. We have:

$$
\begin{aligned}
A(x) & =\frac{\left(x^{4}-x\right)-2005 x^{2}(x-1)}{x-1}=\frac{x\left(x^{3}-1\right)-2005 x^{2}(x-1)}{x-1} \\
& =\frac{(x-1)\left[x\left(x^{2}+x+1\right)-2005 x^{2}\right]}{x-1}=x^{3}+x^{2}+x-2005 x^{2}
\end{aligned}
$$

Then

$$
A(2005)=2005^{3}+2005^{2}+2005-2005^{3}=2005(2005+1)=4022030
$$