Problem 9.5. On an island, there live knights who always tell the truth, and liars who always lie.

One day, 80 residents of the island gathered, each wearing a T-shirt with a number from 1 to 80 (different residents had different numbers). Each of them said one of two phrases:

- "Among those gathered, at least 5 liars have a T-shirt number greater than mine."
- "Among those gathered, at least 5 liars have a T-shirt number less than mine."

What is the smallest number of knights that could have been among these 80 residents?
Answer: 70.

Solution. Suppose there are at least 11 liars. Arrange the numbers on their T-shirts in ascending order and select the liar with the 6th number. Then he must be telling the truth, as there are at least 5 liars with a smaller number and at least 5 liars with a larger number. Thus, there are no more than 10 liars, i.e., there are at least 70 knights.

We will show that there could be exactly 70 knights. For example, let the knights be in T-shirts numbered $1-70$, and the liars in T-shirts numbered $71-80$. All knights and liars with numbers 76 - 80 said the first phrase, while liars with numbers $71-75$ said the second phrase. It is clear that all conditions of the problem are satisfied.