6. On the shore of a round island called Somewhere, there are 20 villages, each inhabited by 20 wrestlers. A tournament was held where each wrestler faced all wrestlers from all other villages. Village A is considered stronger than Village B if at least $k$ matches between wrestlers from these villages end with a victory for a wrestler from Village A. It turned out that each village is stronger than the next one in the clockwise direction. What is the maximum value that $k$ can have? (All wrestlers have different strengths, and the stronger wrestler always wins in a match.)
6. Let's provide an example showing that the described situation is possible when $k \leqslant 290$. We will order all the wrestlers by strength and renumber them in ascending order of strength (the first being the weakest). We will call the 210 weakest wrestlers novices, and the 190 strongest - masters. In particular, any novice will be weaker than any master. We will number the villages counterclockwise. We will place in the first village one of the weakest novices and 19 of the weakest masters; in the second village - two of the weakest remaining novices and 18 of the weakest remaining masters; in the third village - three of the weakest remaining novices and 17 of the weakest remaining masters, and so on; in the last village, we will place the 20 strongest novices. This placement in the villages is shown in the table (in the "Wrestlers" column, numbers in regular font indicate the strengths of masters, and numbers in italics indicate the strengths of novices).

| Villages | Wrestlers | Villages | Wrestlers |
| :---: | :--- | :---: | :--- |
| 1 | $1,211-229$ | 11 | $56-66,356-364$ |
| 2 | $2-3,230-247$ | 12 | $67-78,365-372$ |
| 3 | $4-6,248-264$ | 13 | $79-91,373-379$ |
| 4 | $7-10,265-280$ | 14 | $92-105,380-385$ |
| 5 | $11-15,281-295$ | 15 | $106-120,386-390$ |
| 6 | $16-21,296-309$ | 16 | $121-136,391-394$ |
| 7 | $22-28,310-322$ | 17 | $137-153,395-397$ |
| 8 | $29-36,323-334$ | 18 | $154-171,398-399$ |
| 9 | $37-45,335-345$ | 19 | $172-190,400$ |
| 10 | $46-55,346-355$ | 20 | $191-210$ |

We will show that the $i$-th village is stronger than the $(i-1)$-th village for $i>1$. Indeed, in the $i$-th village, there are $i$ novices and $20-i$ masters. In this case, the masters of the $i$-th village will defeat all in the $(i-1)$-th village, and the novices will defeat the novices, and the total number of victories will be $20(20-i)+i(i-1)=i^{2}-21 i+400$. The vertex of this parabola is at the point $i=10.5$, and the branches are directed upwards, so the minimum value at an integer point is achieved exactly at two values $i=10$ and $i=11$ - and is equal to $10^{2}-21 \cdot 10+400=290$. That is, the $i$-th village is stronger than the $(i-1)$-th village for $k \leqslant 290$. Moreover, the masters of the first village will defeat the novices of the 20th village, and the total number of victories will be $20 \cdot 19=380>290$, i.e., all conditions are met.

We will show that such a situation is impossible when $k>290$. We will order the wrestlers in each village by decreasing strength and select the tenth strongest wrestler in each village. We will show that the village in which the weakest of the selected wrestlers lives cannot be stronger than the next village. Let's denote the selected wrestlers in our and the next village as $A$ and $B$ respectively. Then in our village, there are 11 wrestlers who are not stronger than $A$, and in the next village, there are 10 wrestlers who are at least as strong as $B$. All matches between these wrestlers will end in favor of the next village, and there are 110 such matches, i.e., the number of matches won by a wrestler from our village is no more than $20 \cdot 20-110=290$.