A certain geometric progression has the sum of its first and second terms equal to 20; the sum of its third and fourth terms equal to 20/9. What are these terms?
The task requires solving the following system of equations:

$$
a + aq = 20
$$

$$
aq^2 + aq^3 = \frac{20}{9}
$$

If I divide the second equation by the first, I get:

$$
q^2 = \frac{1}{9}
$$

from which

$$
q_1 = \frac{1}{3}, \quad q_2 = -\frac{1}{3}
$$

Substituting these values into (1), I get the values of \(a\):

$$
\begin{aligned}
& a_1 = \frac{20}{1 + q_1} = \frac{20}{\frac{4}{3}} = 15 \\
& a_2 = \frac{20}{1 + q_2} = \frac{20}{\frac{2}{3}} = 30
\end{aligned}
$$

From this, the four consecutive terms of the geometric progression are:

$1^{\circ} . \quad 15, 5, \frac{5}{3}, \frac{5}{9}$

$2^{\circ} . \quad 30, -10, \frac{10}{3}, -\frac{10}{9}$

(Kiss Béla, Student at the Ludovika Academy, Budapest.)

The problem was also solved by: Ambrus József, Kaposvár; Berényi Győző, Székesfehérvár; *) Friedmann Bernát, S.-A.-Ujhely; Geist Emil, Győr; Grünhut Béla, Pécs; Kohn Márkus, Pécs; Porde Gyula, Szamos-Ujvár; Pósch Gyula, Budapest; Schwartz Béla, Pécs; Visnya Aladár, Pécs;