5. In one class, there are blondes and brunettes. Light-haired students lie in response to any question, while those with dark hair tell the truth. But some children love to dye their hair and dye it every day to the opposite color! (While others never dye their hair). On Monday, it was Petya's birthday, and all the children were asked: "Were you born this month?" and all 20 students gave an affirmative answer. On Friday, the children were asked again, "Were you born this month?" and only 10 students gave an affirmative answer. After the weekend, when asked again on Monday, "Were you born this month?" no one gave an affirmative answer. On Friday, they counted whether there were more blondes or brunettes in the class now. What answer did they get?
5. Answer. There are more brunettes. Solution. On Friday, the children have the same hair color as on Monday, so the answers could only change if the month changed. Therefore, the month definitely did not change from Friday to Monday. Only the children who dye their hair could have changed their answer from Friday to Monday! So there are 10 such children.

Consider those who were dyed brunettes on the first Monday. Since they answered affirmatively, they were born in the $1^{st}$ month. And when they were asked the same question on the next Monday, they should have given an affirmative answer again (since they were blondes at that time). Therefore, there are no such children. This means that all those who dye their hair (10 people) were blondes on the first Monday. By the second Friday, they will become brunettes. It was Petia's birthday the first time. He could not have been a dyed brunette (there are none), so he was a natural brunette. That is, by the second Friday, there are dyed brunettes (10) and at least 1 natural brunette. There are more brunettes.

Grading criteria. Full solution - 7 points. Obtained the result that the month changed - $\mathbf{1}$ point. Obtained the result that 10 people dye their hair - $\mathbf{1}$ point. Obtained the result that all those who dye their hair dye it the same way - $\mathbf{1}$ point. Points for progress are cumulative.