6. In the village of Znoynoye, there are exactly 1000 residents, which exceeds the average population of villages in the valley by 90 people.

How many residents are there in the village of Raduzhny, which is also located in Sunny Valley?
Solution. Let $x$ be the total number of residents in all other villages except Znoynoye. Then the average population is

$$
\frac{1000+x}{10}=100+0.1 x=910, \text{ hence } x=8100
$$

Thus, the average population in 9 villages, except Znoynoye, is 900 people. If there are more than 900 residents in Raduzhny, there must be a village with fewer than 900 residents, but then the number of residents in Raduzhny would differ from the number of residents in Znoynoye by more than 100. This is a contradiction. Therefore, there are no more than 900 residents in Raduzhny. It is obvious that there are no fewer than 900 residents in Raduzhny. Thus, there are exactly 900 residents.

Answer: 900.