The tournament of ties

1. Also, they all end up with the same odd score p, so: 5p=30-e.
2. Then: $30-e=25 \Rightarrow e=5$.
3. With 5 teams, the total number of matches is: $\binom{5}{2}=10$.

Back to problem
Answer: there were exactly 5 ties.
With 5 teams, the total number of matches is:

$$ \binom{5}{2}=10. $$

If there were no ties, they would be distributed:

$$ 10\times3=30 $$

points.
Each tie distributes 2 points instead of 3, so each tie subtracts 1 from the overall total.
If $e$ is the number of ties:

$$ \text{puntos totales}=30-e. $$

Also, they all end up with the same odd score $p$, so:

$$ 5p=30-e. $$

The total points must be between 20 and 30, and be an odd multiple of 5. The only possible value is 25.
So:

$$ 30-e=25 \Rightarrow e=5. $$