Laiko 0.

Around a round table, there is a company of thirty people. Each of them is either a fool or smart. Everyone sitting is asked: Is your right neighbor smart or a fool? In response, a smart person tells the truth, while a fool can say either the truth or a lie. It is known that the number of fools does not exceed $F$. For what maximum value of $F$ can one always, knowing these answers, point to a smart person in this company?
When $F=8$. If $F=0$, then one can point to any person sitting at the table. Let now $F \neq 0$. We divide all those sitting at the table into non-empty groups of consecutive smart and consecutive foolish people; the number of these groups is denoted by $2k$ ($k$ groups of smart people and $k$ groups of foolish people). The number of people in the $i$-th group of smart people is denoted by $w_i$, and the number of people in the $i$-th group of foolish people is denoted by $f_i$ ($1 \leq i \leq k$). Then $f_1 + f_2 + \cdots + f_k \leq F$. Consider the sequence of consecutive answers "smart" and the last person $x$ who is referred to as such. A group of $w_i$ smart people gives such a sequence of length not less than $w_i - 1$, and in this case, $x$ is indeed smart. If, however, $x$ is a fool and is in the $i$-th group of fools, then the length of such a sequence is no more than $f_i - 1$. Therefore, if $\max_i w_i > \max_i f_i$, then one can assert that the last person named smart in the longest sequence of answers "smart" is indeed smart. Since $\max_i w_i \geq \frac{30 - (f_1 + \cdots + f_k)}{k} \geq \frac{30 - F'}{k}$,

$$
\max_i f_i \leq (f_1 + \cdots + f_k) - k + 1 \leq F - k + 1,
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_585f382975b62b8f3818g-11.jpg?height=86&width=1915&top_left_y=2458&top_left_x=0)
sitting at the table. This inequality is equivalent to:

$$
k^2 - (F + 1)k + 30 - F > 0
$$

It holds for all $k$ if $D = (F + 1)^2 + 4(F - 30) < 0$, i.e., when $F < -3 + \sqrt{128} < -3 + 12 = 9$. Thus, when $F \leq 8$, one can identify a smart person based on the given answers. When $F = 9$, this is not always possible. Indeed, consider a company sitting at the table as shown in Figure 104 (the answers are given next to the arrows: s - smart, f - fool; fools are represented by shaded circles). We will rotate this picture around the center by angles of $60^\circ, 120^\circ, 180^\circ, 240^\circ$, and finally $300^\circ$ clockwise. In this case, as one can easily verify, any place can be occupied by either a smart person or a fool, and the sequence of answers remains the same. Therefore, in such a company, it is impossible to identify a smart person based on the given answers.

## Answer

When $F=8$.