A straight quadrilateral pyramid has a square base; each of the side edges is equal to the diagonal of the base. What is the surface area and volume of the pyramid if the diagonal of the base is $d$?
The base diagonal of the pyramid

$$
a=\frac{d}{\sqrt{2}}
$$

its height

$$
M=\sqrt{d^{2}-\frac{d^{2}}{4}}=\frac{d}{2} \sqrt{3}
$$

and the height of its lateral face

$$
m=\sqrt{d^{2}-\frac{a^{2}}{4}}=\sqrt{d^{2}-\frac{d^{2}}{8}}=\frac{d}{2} \sqrt{\frac{7}{2}}
$$

Thus, the surface area of the pyramid

$$
F=\frac{d^{2}}{2}+4 \frac{d}{\sqrt{2}} \cdot \frac{d}{4} \sqrt{\frac{7}{2}}=\frac{d^{2}}{2}(1+\sqrt{7})
$$

and its volume

$$
K=\frac{d^{2}}{2} \cdot \frac{d}{2} \cdot \frac{\sqrt{3}}{3}=\frac{d^{3}}{12} \sqrt{3}
$$

(Miklós Dénes, Budapest.)

The problem was also solved by: A. Baján, E. Bauer, P. Breuer, I. Czigler, V. Erdős, E. Fried, Gy. Grünwald, B. Koffler, F. Sichermann, L. Spitzer, L. Szántó, V. Szilárd, I. Szóbel, A. Vilcsek, R. Viola, Gy. Weinherger.