6. Next to the number 2022 on the board, an unknown positive number less than 2022 was written. Then one of the numbers on the board was replaced by their arithmetic mean. This replacement was performed 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers originally written on the board.
Solution. Let at some point the numbers $a$ and $b$ be written on the board, with $a > b$. Then notice that after the specified operation, the difference between the numbers will become twice as small, regardless of which number we erase, since

$$
a - b = 2\left(a - \frac{a + b}{2}\right) = 2\left(\frac{a + b}{2} - b\right)
$$

Then after 10 operations, we have $2022 - x = 2^{10} * \Delta$, where $x$ is the unknown number written, and $\Delta$ is the last difference, so $x = 2022 - 1024 \Delta$,
since $x$ is positive, the only possible natural value for $\Delta$ is 1, hence $x = 998$.

Answer. 998.

Recommendations for checking. Only the correct answer - 1 point. The correct answer with the assumption that the largest or smallest number was erased each time - no more than 2 points. Formula (1) or a proven similar statement - no less than 3 points. Without justification, assume that the last difference is 1, deduct 2 points.