A circular sector has a perimeter $u$; how long should the radius be to maximize the area of the circular sector, and what is this maximum area? (The problem should be solved 1. in general and 2. if $u=8 \mathrm{~m}$.)
The arc of the circular segment is $i$, the radius of the circle is $r$, so

$$
u=i+2 r
$$

from which

$$
i=u-2 r
$$

substituting this into the formula for the area of the circular segment:

$$
t=\frac{i r}{2}=\frac{u r}{2}-r^{2}
$$

The area of the circular segment will be the largest when

$$
r=\frac{u}{4}=2 \text{ m}
$$

and in this case

$$
t=\frac{u^{2}}{16}=4 \text{ m}^{2}
$$

(Lajos Kertész.)

The problem was also solved by: D. Barna, F. Beck, F. Bojedain, K. Détshy, J. Fekete, K. Goldziher, T. Kármán, H. Schiffer, Ö. Spitzer, I. Szabó, K. Szabó.