For the equation $x^{2}-3 x+q=0$, determine $q$ so that the equation has two equal roots.
If the two roots are equal, then

$$
x_{1}+x_{2}=2 x_{1}=3
$$

thus

$$
x_{1}=x_{2}=\frac{3}{2}
$$

and so

$$
q=x_{1} x_{2}=\frac{9}{4}
$$

(Frigyes Frankel, Budapest.)

Number of solutions: 36.