4. In a football match, the winning team receives 3 points, and the losing team receives 0 points. If the match ends in a draw, each team receives 1 point. With these scoring rules, a tournament is organized in which four teams participate, each team playing two matches with each of the other three. At the end, a ranking is made based on the points accumulated by each team.

a) Determine the total number of matches played in this tournament;

b) If the total sum of all points accumulated by the 4 teams is 34 points, determine the number of matches that ended in a draw.

## NOTE

- All subjects are mandatory;
- Each subject is graded out of 7 points;
- No points are awarded by default;
- The actual working time is 2 hours from the moment the subject is received.


## Mathematics Olympiad

Zone Phase - February 16, 2013

## 5th Grade - grading rubric
4. a) Each team plays 6 matches, so in total there are (6*4):2=12 matches. 2p

b) The total points per match are 3 points when a team wins and 2 points for a draw.

$2 p$

A point is lost from the maximum possible in a draw.

$1 p$

The maximum score is 36, so 2 points were lost, meaning there were 2 draws.

## NOTE

- Any correct solution is graded accordingly.