5. What is the sum of all digits of the number

$$
10^{1}-10^{2}+10^{3}-10^{4}+10^{5}-\cdots-10^{2022}+10^{2023} ?
$$

## Tasks worth 10 points:
5. What is the sum of all digits of the number

$$
10^{1}-10^{2}+10^{3}-10^{4}+10^{5}-\cdots-10^{2022}+10^{2023} ?
$$

## Solution.

| $10^{1}-10^{2}+10^{3}-10^{4}+10^{5}-\cdots-10^{2022}+10^{2023}=$ |  |
| :--- | ---: |
| $=10^{1}+\left(10^{3}-10^{2}\right)+\left(10^{5}-10^{4}\right)+\cdots+\left(10^{2023}-10^{2022}\right)$ | 1 POINT |
| $=10+900+90000+\cdots+9 \cdot 10^{2022}=$ | 1 POINT |
| $=909090 \ldots 90910$ | 1 POINT |
| This number has 2023 digits. | 1 POINT |
| Of these, 1011 digits are 9, 1011 digits are 0, and one is a 1 (the second-to-last digit). | 1 POINT |
| The sum of all digits of this number is |  |
| $1011 \cdot 9+1=9099+1=9100$. | 1 POINT |
| .......................................................................................... |  |