5. In 1685, having found himself on an uninhabited island, the native Friday met Robinson Crusoe on the very first day. Good spirits taught Friday the European calendar and hinted that only one day a year, February 13, Robinson tells the truth, while on other days he lies. Every day, Friday asks Robinson Crusoe one question of the form "Is it true that today is such-and-such a date of such-and-such a month?" Will Friday be able to find out the date of his arrival on the uninhabited island within 183 days?
Answer: Yes. Let's number the days of the year from the zeroth (February 13) to the 364th. Mark the 183rd day on the calendar, and call it "day $X$". Between the zeroth day and day $X$, 182 days pass, and between day $X$ and the zeroth day, 181 days pass.

Suppose Friday asks for the first 181 times whether today is day $X$. As soon as he hears the answer "no," it means that it is either the zeroth day or day $X$. Then, with the next question, he can find out which of these two days was yesterday (for example, by asking about February 14). If, however, Friday gets the answer "yes" 182 times in a row, it means that 182 times in a row it was neither the zeroth day nor day $X$, which is only possible if he started asking questions on February 14.