6.1. How many triangles with integer sides have a perimeter equal to 27? (Triangles that differ only in the order of the sides - for example, $7,10,10$ and $10,10,7$ - are considered the same triangle.)
Answer: 19.

Solution: Arrange the sides in ascending order: $a \leqslant b \leqslant c$. Then the smaller side $a$ does not exceed 9. If $a=1$ or $a=2$, there is one such triangle. If $a=3$ or $a=4$, there are two such triangles. If $a=5$ or $a=6$, there are three such triangles. If $a=7$, there are four such triangles. If $a=8$, there are two such triangles. If $a=9$, there is one such triangle. In total, we get 19 triangles.