30 teams are taking part in a football championship. Prove that at any moment in the contest there will be two teams who have played the same number of matches up to that moment, assuming every team plays every other team exactly once by the end of the tournament.
Six chess players participated in a tournament. Each two participants of the tournament played one game against each other. How many games were played? How many games did each participant play? How many points did the chess players collect all together?