Question 174
Along the coast of a circular island there are 20 different villages. Each of these villages has 20 fighters, and all 400 fighters have different strengths.
Each pair of neighbouring villages $A$ and $B$ now holds a contest in which each of the 20 fighters of village $A$ competes against each of the 20 fighters of village $B$. In each match the stronger fighter wins. We say that village $A$ is stronger than village $B$ if a fighter of village $A$ wins in at least $k$ of the 400 matches.
It turns out that every village is stronger than its clockwise neighbouring village. Determine the maximum value of $k$ for which this can be the case.
Each pair of neighbouring villages $A$ and $B$ now holds a contest in which each of the 20 fighters of village $A$ competes against each of the 20 fighters of village $B$. In each match the stronger fighter wins. We say that village $A$ is stronger than village $B$ if a fighter of village $A$ wins in at least $k$ of the 400 matches.
It turns out that every village is stronger than its clockwise neighbouring village. Determine the maximum value of $k$ for which this can be the case.