15. Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tournament the total number of points gained by all the teams was 61 . What is the maximum number of points that the tournament's winning team could have obtained?
A 16
B 17
C 18
D 19
E 21
SolUTION
B

Let $D$ be the number of drawn matches; then the number of points awarded for draws is $2 D$ (one point for each team in the match). Let $W$ be the number of matches that resulted in a win; then the number of points awarded for these matches is $3 W$. Thus the total number of points is $2 D+3 W=61[1]$. Each of the 8 teams played 7 others, so the number of matches is $8 \times 7 \div 2=28$ (each match has been counted twice, so we need to divide by 2 ). Since each match is either a draw or a win (for one team), we have $D+W=28$ [2].
Equation [1] - $2 \times$ equation [2] gives $W=5$. So exactly 5 matches were won. Hence the maximum number of points that the winning team could have obtained is $5 \times 3+2=17$ ( 5 wins and 2 draws).