4. In city $\mathrm{N}$, only blondes and brunettes live. Blondes always lie, while brunettes always tell the truth. Every day, the residents dye their hair the opposite color. On one Monday in October, everyone born in the fall was asked: "Were you born this month?" - and 200 residents answered "yes," while no one answered "no." On Friday of the same week, everyone born in the fall was asked again: "Were you born this month?" - and this time only 50 of them gave an affirmative answer. How many "yes" answers were given to the same question on the last Monday of September, if there were exactly four Mondays in October of that year and no one in the city $\mathrm{N}$ was born in November?
4. Answer. 0.

Solution. Note that on Monday and Friday of the same week, the residents have the same hair color: Monday - Tuesday - Wednesday - Thursday - Friday. Then, if the same people answer differently on Monday and Friday, it means that the month has changed, and the brunettes will answer "no" to the same question.

If the month has changed, then it is the last Monday of October. Since there were 4 Mondays in October, the residents will have the same hair color again on the last Monday of September: M (September) - M1 - M2 - M3 - M4 (October) (since the hair color changes each subsequent Monday). The first time all brunettes answered "yes," it means that all residents who were brunettes on that day were born in October. On the following Friday and the last Monday of September, they will also be brunettes, but since the month will change, all of them will answer "no."

Let's pay attention to the blondes. The first time all of them answered "yes," it means that none of them were born in October. We also know that no one was born in November. Therefore, all blondes were born in September and will also answer "no" on the last Monday of September.

Thus, on the last Monday of September, 0 residents will answer "yes."

## Grading Criteria:

Correct and justified solution - 7 points. In all other cases - $\mathbf{0}$ points.