2. (3 points) On the Island of Misfortune, there live knights who always tell the truth, and liars who always lie. One day, $n$ islanders gathered in a room.

The first one said: "Exactly 1 percent of those present in this room are liars."

The second one said: "Exactly 2 percent of those present in this room are liars."

and so on

The person with number $n$ said: "Exactly $n$ percent of those present in this room are liars."

How many people could have been in the room, given that it is known for sure that at least one of them is a knight?
Answer: 100

## Solution:

There must be exactly one knight, as all other islanders in the room contradict each other. This means that the knights are $\frac{1}{n}$ of the total number of people. Let them be $k$ percent of the total number of people present, which means the person with the number $100-k$ is telling the truth.

We get that $\frac{1}{n}=\frac{k}{100}$, where $100-k \leqslant n$.

This can be rewritten as $n k=100$ and $k+n \geqslant 100$, from which $n k-k-n \leqslant 0$.

Therefore, $n-k-n+1 \leqslant 1$, that is, $(n-1)(k-1) \leqslant 1$. Since $n$ and $k$ are both natural numbers, we get that either one of them is 1, or both are 2. The latter case does not satisfy the equation $n k=100$.

If $n=1$, we get that the only person is a liar, which contradicts the condition.

If $k=1$, then $n=100$ and the first person is telling the truth. This is the answer.