[Mathematical logic (miscellaneous).]

On an island, there live liars and knights, a total of 2001 people. Knights always tell the truth, while liars lie. Each resident of the island stated: "Among the remaining residents of the island, more than half are liars." How many liars are there on the island?

#
Take one knight and one liar. From the statements made by them, lower and upper bounds on the number of liars can be derived.

## Solution

All the inhabitants of the island cannot be liars, otherwise each of them would be telling the truth. Let's take some knight. From his statement, it follows that the number of liars on the island is greater than (2001-1)/2=1000. Now let's take some liar. His statement is false, so apart from him, no more than half of the island's inhabitants are liars. This means that apart from him, there are no more than 2000/2=1000 liars on the island, i.e., together with him, there are no more than 1001 liars. Thus, from the obtained bounds on the number of liars, we get that the only possibility is when there are exactly 1001 liars on the island.

## Answer

The problem