8.4. On the Island of Liars and Knights, a circular arrangement is called correct if each person standing in the circle can say that among their two neighbors, there is a representative of their tribe. Once, 2019 natives formed a correct circular arrangement. A liar approached them and said: "Now we can also form a correct circular arrangement together." How many knights could there have been in the initial arrangement?
Answer: 1346.

Solution. We will prove that a correct arrangement in a circle is possible if and only if the number of knights is at least twice the number of liars.

Indeed, from the problem's condition, it follows that in such an arrangement, each liar has two knights as neighbors, and among the neighbors of any knight, there is at least one knight. Therefore, the correct arrangement should look like this: a group of knights, a liar, a group of knights, a liar, and so on (with at least two knights in each group). This means that in such an arrangement, the number of knights is at least twice the number of liars.

In the reverse direction: if the number of knights is at least twice the number of liars, then we can make an arrangement of the form RRKRRK..., and place any remaining knights (if they exist) between any two knights. Thus, if this condition is met, a correct arrangement is possible.

Let \( P \) be the number of knights and \( L \) be the number of liars in the correct arrangement specified in the problem, then \( P \geq 2L \). The liar who arrived told a lie, so together with him, a correct arrangement is impossible, hence \( P \leq 2L + 1 \). Therefore, \( P = 2L \) or \( P = 2L + 1 \). In the first case, in the initial arrangement, \( 2019 \cdot \frac{2}{3} = 1346 \) knights, and the second case is impossible because the number \( (2019 - 1) \cdot \frac{2}{3} \) would not be an integer.

Evaluation criteria.

«+» A complete and justified solution is provided.

«士» A generally correct reasoning is provided, containing minor gaps or inaccuracies (for example, only the necessary condition for the existence of a correct arrangement is proven or a computational error is made at the end).

«Ғ» The answer is obtained based on the assumption that the number of knights is twice the number of liars, but this is not proven.

«-» Only the answer is provided.

«-» The problem is not solved or is solved incorrectly.