7.1. In a football tournament, 20 teams participated. Each team played one match against each other. A team receives 3 points for a win, 1 point for a draw, and 0 points for a loss. After the tournament, Vasya calculated the total points scored by the teams. It turned out to be 500. How many matches ended in a draw?
Answer: 70

Solution. Each team played 19 matches, so the total number of matches played was $20 \cdot 19 / 2=190$ (we divide by two because each match is counted twice). If there were no draws, the total points would be $3 \cdot 190=570$. Each draw results in one point less in the total compared to a match that ends with one team winning (the total points scored is 2, not 3). Since we need to reduce the total by 70 points, there must be 70 draws.