A right-angled triangle has a hypotenuse of $37 \mathrm{~cm}$; the area of the triangle does not change if one of the legs is increased by $7 \mathrm{~cm}$ and the other leg is decreased by $2 \mathrm{~cm}$. What are the lengths of the legs?
Let the legs be $a$ and $b$, then

$$
(a+7)(b-2)=a b \quad \text { and } \quad a^{2}+b^{2}=37^{2}
$$

which has positive roots: $a=35, \quad b=12$.

Number of solutions: 32.