Let $x, x^{\lg x}, y^{\lg y}, (xy)^{\lg xy}$ be consecutive terms of a geometric sequence. Write down these four numbers.
In a geometric sequence, when the elements are written as powers of a common base, the exponents form an arithmetic sequence. In our case, the common base is 10. Thus, the elements of the sequence are:

$$
10^{\lg x}, 10^{(\lg x)^{2}}, 10^{(\lg y)^{2}}, 10^{(\lg x y)^{2}}
$$

The difference between the exponents of consecutive terms is equal:

$$
(\lg x)^{2}-\lg x=(\lg y)^{2}-(\lg x)^{2}=(\lg x y)^{2}-(\lg y)^{2}
$$

From the equality of the first and last differences,

$$
(\lg x-1) \lg x=(\lg x y-\lg y)(\lg x y+\lg y)=(\lg x) \cdot(\lg x+2 \lg y)
$$

Subtracting the right side from the left, we get

$$
(\lg x)(1+2 \lg y)=0
$$

Thus, either $\lg x=0$ or $\lg y=-1 / 2$.

I. In the first case, from the equality of the first two differences in (1),

$$
(\lg y)^{2}=0, \lg y=0
$$

In this case, $x=y=1$, and indeed all four numbers are 1.

II. If $\lg y=-1 / 2$, then again from the equality of the first two differences in (1),

$$
\begin{gathered}
2(\lg x)^{2}-\lg x-\frac{1}{4}=0 \\
\lg x=\frac{1 \pm \sqrt{1+2}}{4}=\left\{\begin{array}{c}
\frac{1+\sqrt{3}}{4} \\
-\frac{\sqrt{3}-1}{4}
\end{array}\right.
\end{gathered}
$$

The two sets of values for the exponents of the 2nd, 3rd, and 4th terms are

$$
\begin{gathered}
\left(\lg x_{1}\right)^{2}=\frac{2+\sqrt{3}}{8} \\
(\lg y)^{2}=\frac{1}{4}, \quad\left(\lg x_{1} y\right)^{2}=\left(\frac{1+\sqrt{3}}{4}-\frac{1}{2}\right)^{2}=\left(\frac{\sqrt{3}-1}{4}\right)^{2}=\frac{2-\sqrt{3}}{8} \\
\left(\lg x_{2}\right)^{2}=\frac{2-\sqrt{3}}{8} \\
\left(\lg x_{2} y\right)^{2}=\left(-\frac{\sqrt{3}-1}{4}-\frac{1}{2}\right)^{2}=\left(-\frac{\sqrt{3}+1}{4}\right)^{2}=\frac{2+\sqrt{3}}{8}
\end{gathered}
$$

From this, calculating $x$ and $y$,

$$
\begin{gathered}
x_{1}=10^{\frac{1+\sqrt{3}}{4}} \approx 10^{0.6830} \approx 4.820 ; \quad x_{2}=\frac{1}{10^{\frac{\sqrt{3}-1}{4}}} \approx 10^{0.8170-1} \approx 0.6561 \\
y=\frac{1}{\sqrt{10}} \approx 0.3162
\end{gathered}
$$

The following values were obtained for the 4 elements of the sequence:

|  | $10^{\lg x}$ | $10^{(\lg x)^{2}}$ | $10^{(\lg y)^{2}}$ | $10^{(\lg x y)^{2}}$ |
| :---: | :---: | :---: | :---: | :---: |
| $\mathrm{I}$ | 1 | 1 | 1 | 1 |
| $\mathrm{II}_{1}$ | 4.820 | $10^{\frac{2+\sqrt{3}}{8}} \approx 2.927$ | $10^{\frac{1}{4}} \approx 1.778$ | $10^{\frac{2-\sqrt{3}}{8}} \approx 1.080$ |
| $\mathrm{II}_{2}$ | 0.6561 | $10^{\frac{2-\sqrt{3}}{8}} \approx 1.080$ | $10^{\frac{1}{4}} \approx 1.778$ | $10^{\frac{2+\sqrt{3}}{8}} \approx 2.927$ |

István Györi (Székesfehérvár, Ságvári E. Mechanical Engineering High School III. grade)