17-202 A scalene triangle, whose sides are integers, has a perimeter less than 13. The number of such triangles is
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(E) 18.
(7th American High School Mathematics Examination, 1956)
[Solution] Let $x, y, z$ be the lengths of the sides of a triangle. According to the problem and the triangle inequality, we have
$$
\left\{\begin{array}{l}
x+y+z \leq 12, \\
y+z>x, \\
z+x>y,
\end{array}\right.
$$

Since $x, y, z$ are all integers, from (1) we have $x+y+z \leq 12$.
Assuming $z$ is the longest side, from (2) we have $2z < 12$, which means $z < 6$.
Also, $z \geq 4$, otherwise if $z=3$, the other two sides would be 1 and 2, which cannot form a triangle.
Therefore, we have
$$
4 \leq z < 6,
$$
When $z=4$, we have $y=3, x=2$;
When $z=5$, we have $y=4, x=3$ or $y=4, x=2$.
From the above analysis, the answer is $(C)$.