16 teams took part in a handball tournament where a victory was worth 2 points, a draw – 1 point and a defeat – 0 points. All teams scored a different number of points, and the team that ranked seventh, scored 21 points. Prove that the winning team drew at least once.
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?
Several football teams are taking part in a football tournament, where each team plays every other team exactly once. Prove that at any point in the tournament there will be two teams who have played exactly the same number of matches up to that point.
The Russian Chess Championship is made up of one round. How many games are played if 18 chess players participate?
In a tournament by the Olympic system (the loser is eliminated), 50 boxers participate. What is the minimum number of matches needed to be conducted in order to identify the winner?