A perpendicular, rectangular sheet of equal thickness has a diagonal of $d$, and its total surface area is $2 a^{2}$. What are the lengths of the sides of the sheet, if they form an arithmetic progression?
If the edges are $b-c, b, b+c$, then

$$
d^{2}=b^{2}-2 b c+c^{2}+b^{2}+b^{2}+2 b c+c^{2}
$$

or

$$
d^{2}=3 b^{2}+2 c^{2}
$$

and

$$
2 a^{2}=2 b(b-c)+2 b(b+c)+2(b-c)(b-c)
$$

or

$(2)$

$$
2 a^{2}=6 b^{2}-2 c^{2}
$$

From (1) and $(2)$

$$
b=\frac{1}{3} \sqrt{d^{2}+2 a^{2}}, \quad c=\sqrt{\frac{d^{2}-a^{2}}{3}}
$$

Therefore, the edges of the equilateral plane are:

$$
\frac{1}{3} \sqrt{d^{2}+2 a^{2}}-\sqrt{\frac{d^{2}-a^{2}}{3}}, \quad \frac{1}{3} \sqrt{d^{2}+2 a^{2}}, \quad \frac{1}{3} \sqrt{d^{2}+2 a^{2}}+\sqrt{\frac{d^{2}-a^{2}}{3}}
$$

(Vilmos Erdös, Budapest.)