2. (10 points) Around a circle, $n$ balls, white and red, are arranged. Among any five consecutive balls, exactly two are white. For which $n$ is this possible?
a) 2021 ;

б) 2022 ;

в) 2023 ;

г) 2024 ;

д) none of the given answers is correct.
Answer: d).

Solution. Let's start numbering the balls in the circle from some white ball. Among the first five balls, by condition, there are exactly two white ones - the first and the $i$-th, where $i$ is one of the numbers from 2 to 5 inclusive. Now consider the group of five balls starting from the second: among the second to the fifth, the only white one is the $i$-th ball; therefore, to have exactly two white balls in this group, the sixth position must also be occupied by a white ball. By similar reasoning, the white balls must be placed at positions numbered $5k+1, k=0,1, \ldots$ and at positions numbered $5k+i, k=0,1, \ldots$. Therefore, the circle with such conditions on the placement of the balls must consist of a number of balls divisible by 5.