# Task 8.5

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 arrangement in a circle. A liar approached them and said: "Now we can also form a correct arrangement in a circle together." How many knights could there have been in the initial arrangement?

## Number of points 7
Answer:

1346

## Solution

We will prove that a correct arrangement around 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, there are at least twice as many knights as liars.

In the reverse direction: if there are at least twice as many knights as 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, there are \( 2019 \cdot \frac{2}{3} = 1346 \) knights, and the second case is impossible because the number \( (2019 - 1) \cdot \frac{2}{3} \) would not be an integer.

Additional criteria for evaluation.

"4 or 5 points: 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)."
"2 or 3 points: The answer is obtained based on the assumption that there are twice as many knights as liars, but this is not proven."