8.5. Girls stood in a circle, some of them wearing scarves. A circle is called correct if each girl without a scarf has a neighbor wearing a scarf. a) What is the minimum number of scarves in a correct circle of 25 girls? b) Prove that if in this correct circle of 25 girls there are more than 12 scarves, then some girls can take off their scarves, and the circle will still be correct.
Answer: a) 9. Solution. a) Note that among three consecutive girls, there is at least one handkerchief: otherwise, the girl in the middle would not have a neighbor with a handkerchief. Fix one girl with a handkerchief in the circle, say Tanya, and consider three girls in sequence clockwise after Tanya. There are a total of 8 such triplets, and in each of them, there is at least one handkerchief. Thus, the total number of handkerchiefs is no less than $1+8=9$, and the estimate is obtained. Now let's construct an example of a correct circle with 9 handkerchiefs: for this, we can put a handkerchief on Tanya, and in each of the specified triplets, on the girl standing in the middle. b) Suppose there are more than 12 handkerchiefs in the circle. Fix a girl without a handkerchief, say Olya (if all 25 girls have handkerchiefs, the situation is obvious: any of them can take off their handkerchief). The remaining 24 girls can be divided into six quartets, counting clockwise after Olya. Then in one of these quartets, there will be at least three handkerchiefs (by the pigeonhole principle or reasoning by contradiction: otherwise, there would be no more than $2 \cdot 6=12$ handkerchiefs in total). Therefore, among the three girls with handkerchiefs in this quartet, the middle girl can obviously take off her handkerchief and the circle will remain correct.