(a) A given number is to be divided into two parts so that the sum, one term of which is the quotient of the first and second parts, and the other term is the reciprocal of this quotient, is minimized.

(b) A given number is to be divided into two parts so that the sum of the square roots of the parts is maximized.
a.) Let the given number be $a$, the parts $x$ and $a-x$, then

$$
y=\frac{x}{a-x}+\frac{a-x}{x}=\frac{x^{2}+(a-x)^{2}}{x(a-x)}=\frac{a^{2}-2 x(a-x)}{x(a-x)}=\frac{a^{2}}{x(a-x)}-2
$$

$y$ is minimized when the product $x(a-x)$ is maximized, but

$$
x+(a-x)=a
$$

and thus constant, therefore $y$ is minimized when $x=\frac{a}{2}$.

(Ferencz Aczél, Budapest.)

Number of solutions: 33.