Among four numbers, the first three form an arithmetic sequence, the second three form a geometric sequence. The sum of the first and fourth number is 37, the sum of the second and third is 36. Which are these four numbers?
I. Solution: Let the four numbers be $x, y, z, t$.

According to the problem:

$$
\begin{array}{r}
x+z=2 y \\
y t=z^{2} \\
x+t=37 \\
y+z=36
\end{array}
$$

Subtracting (4) from (3)

$$
(x-y)+(t-z)=1
$$

Substituting the value of $x-y=y-z$ from (1) into (5)

$$
y+t-2 z=1
$$

Substituting the value of $t=\frac{z^{2}}{y}$ from (2) into (6)

$$
y+\frac{z^{2}}{y}-2 z=1
$$

Finally, substituting the value of $y$ from (4) into (7)

$$
36-z+\frac{z^{2}}{36-z}-2 z=1
$$

Removing the fraction and rearranging (if $z \neq 36$)

$$
4 z^{2}-143 z+1260=0
$$

From which

$$
z_{1}=20, \quad z_{2}=\frac{63}{4}
$$

Based on (4), (1), and (3)

$$
\begin{array}{lll}
y_{1}=16, & x_{1}=12, & t_{1}=25 \\
y_{2}=\frac{81}{4} & x_{2}=\frac{99}{4} & t_{2}=\frac{49}{4}
\end{array}
$$

Thus, the following two sequences of numbers satisfy the requirements of the problem:

$$
12,16,20,25, \quad \text { or } \frac{99}{4}, \quad \frac{81}{4}, \quad \frac{63}{4}, \quad \frac{49}{4}
$$

János Galambos (Veszprém, Lovassy g. II. o. t.)

II. Solution: We can also solve it with two unknowns. If we denote the first term by $a$ and the difference of the arithmetic sequence by $d$, then according to the problem, the four numbers are:

$$
a, \quad a+d, \quad a+2 d, \quad \frac{(a+2 d)^{2}}{a+d}
$$

Furthermore,

$$
\begin{aligned}
a+\frac{(a+2 d)^{2}}{a+d} & =37 \\
2 a+3 d & =36
\end{aligned}
$$

From (9)

$$
a=\frac{36-3 d}{2}
$$

and thus

$$
a+d=\frac{36-d}{2}
$$

and

$$
a+2 d=\frac{36+d}{2}
$$

Removing the fraction in (8) and substituting the values from (10) and (11)

$$
\frac{36-3 d}{2} \cdot \frac{36-d}{2}+\frac{(36-d)^{2}}{4}=37 \frac{36-d}{2}
$$

Rearranging

$$
2 d^{2}+d-36=0
$$

From which

$$
d_{1}=4, \quad d_{2}=-\frac{9}{2}
$$

and thus from (9)

$$
\begin{aligned}
a_{1}=12 \quad a_{2} & =\frac{99}{4} . \\
& \text { Kálmán Árokszállási (Sárospatak, Rákóczi g. III. o. t.) }
\end{aligned}
$$