A right-angled triangle has a hypotenuse of $60 \mathrm{~cm}$, and the sum of the legs is $84 \mathrm{~cm}$; what are the lengths of the legs?
If one leg is $a$, then the other is $84-a$, so

$$
a^{2}+(84-a)^{2}=60^{2}
$$

from which $a_{1}=48, a_{2}=36$. Therefore, one leg is 48, the other is $36 \mathrm{~cm}$.

Number of solutions: 31.