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?
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.