Problem 4. In a football tournament, 10 football teams participated, and each team played exactly one match against every other team. For each win, 3 points are awarded, for a draw, 1 point, and for a loss, 0 points. At the end, a final table is given showing the total points of each team. If the total points in the final table amounted to 120, then how many matches in the football tournament ended in a draw?
Solution. The total number of football matches played in the tournament is $9+8+7+\ldots+2+1=45$. If we denote the number of draws by $x$, then $45-x$ is the number of matches that ended with a winner. From this, we get the equation

$$
3(45-x)+2 x=120
$$

whose solution is $x=15$.

Thus, 15 matches in the football tournament ended in a draw.