Twenty students each play 19 matches of an eliminating game. For each student, a win gets them 1 point and a loss gets them 0 points. True or false: as long as no student wins all 19 matches, then at least 2 students get the same score.
Each student plays 19 matches and none of them wins all 19, so there are 19 possible scores (18 through 0) . Since there are 20 students, at least one score must be achieved by two students.