In the sense of the task:

$$
x: u=v: y
$$

$$
x+y=20
$$

$$
u+v=34
$$

$$
x^{2}+y^{2}+u^{2}+v^{2}=1300
$$

Squaring (2) and (3) and adding them.

$$
x^{2}+y^{2}+2 x y+u^{2}+v^{2}+2 u v=1556
$$

Considering (4) and (1):

$$
4 x y=4 u v=256
$$

or

$$
x y=u v=64
$$

(2), (3), and (5) thus lead to the following two systems of equations.

$$
\begin{aligned}
x+y & =20, & & u+v=34 \\
x y & =64, & & u v=64
\end{aligned}
$$

Solving these systems of equations, we obtain the following values.

$$
x=16,4,16,4 ; u=32,32,2,2 ; y=4,16,4,16 ; \quad v=2,2,32,32
$$

The problem is solved.

Feuer Mór, Friedmann Bernát, Fröhlich Károly, Galter János, Geist Emil, Goldstein Zsigmond, Grünhut Béla, Hofbauer Ervin, Kántor Nándor, Kornis Ödön, Langheim Pál, Reif Jenő, Schneider Béla.
Four numbers form a geometric progression; the sum of the extremes is 20, the sum of the means is 34, the sum of the squares of the numbers is $1300;$ what are these numbers?