Problem 5.8. A boastful fisherman says the same phrase every day: "Today I caught more perch than I did the day before yesterday (2 days ago), but less than I did 9 days ago." What is the maximum number of days in a row that he can tell the truth?
Answer: 8.

Solution. First, let's provide an example where he tells the truth for 8 days in a row. The numbers of perch he caught on consecutive days are indicated:

$$
2020202020202061 \underbrace{728394105}_{\text {truth }} \text {. }
$$

Now let's prove that he could not have told the truth for 9 days in a row. Suppose he did. Let's denote the truthful days as the 3rd, ..., 11th, the previous day as the 2nd, and the day before that as the 1st.

Then on the 10th day, he caught more than on the 8th, which is more than on the 6th, 4th, and 2nd. At the same time, on the 11th day, he caught less than on the 2nd, since it was exactly 9 days ago. Therefore, on the 11th day, he caught less than on the 10th.

On the other hand, on the 11th day, the fisherman caught more than on the 9th, which is more than on the 7th, 5th, 3rd, and 1st. At the same time, on the 10th day, he caught less than on the 1st. This means that on the 10th day, he caught less than on the 11th, which contradicts the previously obtained conclusion.