1. How many units are in the number $S=9+99+999+\cdots+\overbrace{9 \ldots 90}^{1000}$?
$$
\begin{aligned}
& S=10-1+100-1+1000-1+\cdots+10^{1000}-1=10\left(1+10+100+\cdots+10^{999}\right)- \\
& -100=10 \frac{10^{1000}-1}{9}-1000=\underbrace{111 \ldots 10}_{999 \text { ones }}-1000 \Rightarrow
\end{aligned}
$$

the number has 998 ones. Answer: $\{998\}$.