Question 10
Along the coast of an island there are 20 villages. Each village has 20 fighters. Every fighter fights all the fighters from all the other villages. No two fighters have equal strength and the stronger fighter wins the fight. We say that the village $A$ is stronger than the village $B$ if in at least $k$ fights among the fighters from $A$ and $B$ a fighter from the village $A$ wins. It turned out that every village is stronger than its neighbour (in the clockwise direction). Show that the maximal possible $k$ is 290.