Hello billymBy the end of the tournament, each team has played 9 matches. So at any given time, the number of matches that each team has played is between 0 and 9 inclusive. There are 10 teams, and the only way they can all have played different numbers of matches is if all 10 numbers from 0 to 9 are used.

But, since each match is between 2 teams, if we add up the number of matches played by each team, we must get an even number. And 0 + 1 + 2 + ... + 9 = ?

Grandad