The number $k$ should be divided into two parts in such a way that the sum of the square roots of the parts is as large as possible.
Let one part be $x^{2}$, the other $k-x^{2}$. We seek the maximum value of the function

$$
y=x+\sqrt{k-x^{2}}
$$

Move $x$ to the left side, and square both sides of the equation:

$$
\begin{aligned}
& y^{2}-2 y x+x^{2}=k-x^{2} \\
& 2 x^{2}-2 y x+y^{2}-k=0
\end{aligned}
$$

from which

$$
x=\frac{y}{2} \pm \sqrt{\frac{2 k-y^{2}}{4}}
$$

$x$ can only have real values if

$$
y^{2} \leqq 2 k
$$

Thus, the maximum value of $y^{2}$ is $2 k$, and so the maximum value of $y$ is $\sqrt{2 k}$. At this maximum value of $y$, $x=\sqrt{\frac{k}{2}}$, and thus one part is $x^{2}=\frac{k}{2}$, the other $k-x^{2}=\frac{k}{2}$.

The sum of the square roots is therefore the largest when the parts are equal.

The problem was solved by: Friedmann Bernát, S.-A.-Ujhely; Galter János, Székely-Udvarhely; Geiszt Emil, Győr; Goldstein Zsigmond, Nyíregyháza; Grünhut Béla, Pécs; Hofbauer Ervin and Kántor Nándor, Budapest; Szabó István, Debreczen; Visnya Aladár, Pécs; Zemplén Gyóző, Fiume.