Adva van két szám, találj ki olyan harmadik számot, hogy ha kettőnek-kettőnek összegét a megmaradt harmadikkal megszorzom, olyan három számot kapjak, melyek között ugyanazon különbség áll fenn.

Given are two numbers, find a third number such that if I multiply the sum of the two by the remaining third, I get three numbers between which the same difference exists.
Let the given numbers be $a$ and $b$; let $a < b$.

The solution to the problem must be discussed in three cases according to the assumption about the sought number, whether it is less than $a$, between $a$ and $b$, or greater than $b$.

1. $x < a < b$, in this case, according to the meaning of the problem, we get the following three numbers: $b x + a b$, $a x + a b$, and $a x + b x$, which follow each other in order of their magnitude.

The condition gives the following equation:

$$
b x - a x = a b - b x
$$

from which

$$
x = \frac{a b}{2 b - a}
$$

2. $a < x < b$, then by multiplying two and two of their sums, we get the numbers $a b + b x$, $a x + b x$, and $a x + a b$, which again follow each other in order of their magnitude; from the condition

$$
\begin{gathered}
a b - a x = b x - a b \\
x = \frac{2 a b}{a + b} .
\end{gathered}
$$

3. $a < b < x$, then the numbers are $a x + b x$, $a b + b x$, and $a b + a x$, which again follow in order of their magnitude; from the condition

$$
\begin{gathered}
a x - a b = b x - a x \\
x = \frac{a b}{2 a - b} .
\end{gathered}
$$

In this last case, $x = \infty$ if $2 a = b$.