The geometric mean of two numbers is 600, their arithmetic mean is 49 greater than their harmonic mean. What are these numbers?
According to the task

$$
x y=360000
$$

$$
\frac{x+y}{2}-49=\frac{2 x y}{x+y}
$$

(2) can also be written as

$$
(x+y)^{2}-98(x+y)=1440000
$$

from which

$$
x+y=1250, \text { or } x+y=-1152 \text {. }
$$

Connecting the first equation (1), we get:

$$
x_{1}=y_{2}=800, x_{2}=y_{1}=450
$$

The system of equations $x+y=-1152$ and $x y=360000$ does not have a solution in real numbers.

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The problem was also solved by: Harsányi Z., Jánosy Gy., Pichler S., Reich Z., Sárközy E., Schlesinger Gy., Sehwarz Gy., Szóke D., Tóth B.