The first term of a certain arithmetic and geometric progression is 5; the second term of the arithmetic progression is 2 less than the second term of the geometric progression; the third term of the geometric progression is equal to the sixth term of the arithmetic progression. What progressions satisfy the conditions given here?
The first terms of the geometric progression are: $5, 5q, 5q^2$; the first term of the arithmetic progression is 5, the second term is $5q-2$; thus the difference is $5q-7$ and so the sixth term is $25q-30$. Therefore, according to the problem:

$$
5q^2 = 25q - 30
$$

or

$$
q^2 - 5q + 6 = 0
$$

from which

$$
q = 3 \quad \text{and} \quad q' = 2.
$$

Thus, the problem has two corresponding progressions:

the geometric progressions: $5, 15, 45, \ldots ; 5, 10, 20, \ldots$

the arithmetic progressions: $5, 13, 21, \ldots ; 5, 8, 11, \ldots$

(Daniel Barna, Budapest, ref. fógymn.)

Number of solutions: 41.