A round-robin tournament among $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?
There is a solution by applying Hall's theorem, but I was wondering if there exists an elementary solution that someone without knowledge of Hall's theorem could find during the test.