10.1. Let's call the following 2 operations, each containing 2 arithmetic actions, festive: for the first festive operation, the number is halved (if it is even), then increased by 1, for the second - the number is multiplied by 5, then increased by 1. Can we obtain the number 2022 from the number 2023 by performing only festive operations

#
# Solution:

For the second operation, you can get a number where the units digit is 1 or 6. Therefore, 2022 can only be obtained by the first operation.

$10 x+2=\frac{y}{2}+1$, where $x$ is some integer and $y$ is the number obtained before applying the first operation. We get $y=20 x+2$. Thus, it can be stated that after applying the first operation, a number with 2 in the units place can only come from a number with 2 in the units place. However, it cannot result from the first operation.

2023 is an odd number, so we need to apply both operations.

Answer: No, it cannot.

| Criteria | Points |
| :---: | :---: |
| 1. Complete solution of the problem | 7 |
| 2. Generally correct solution. It is stated but not explained why a 2 in the units place can only result from a number with 2 in the units place after the first operation. | 6 |
| 3. Both statements about the last digit are mentioned, but the contradiction is not shown. | 2 |
| 4. One of the statements about the last digit is mentioned. | 1 |
| 5. Incorrect solution. | 0 |