3. At a large round table, 100 people are sitting. Each of them is either a knight, a liar, or a fool. A knight always tells the truth, a liar always lies. A fool tells the truth if a liar is sitting to their left; lies if a knight is sitting to their left; and can say anything if a fool is sitting to their left. Each of them said, "A liar is sitting to my right." How many liars are there in total? List all possible answers and prove that there are no others.

(V. Migrin)
3. Answer: 50 or 0 liars. The first scenario is realized when knights and liars sit alternately around the table. The second scenario is when all those sitting at the table are eccentrics who tell lies.

Let's prove that there cannot be a different number of liars at the table. Indeed, if there is a liar at the table, then to his right sits an eccentric or a knight. In this situation, both of them will tell the truth, which means the next one after them is again a liar. We get that liars and non-liars alternate, i.e., liars make up exactly half.

If there are no liars at the table, then there are no knights either (because to the right of a knight there must be a liar) and at the table are only eccentrics.