Solve the following system of equations:

$$
\begin{gathered}
x+y+z=\frac{7}{2} \\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{7}{2} \\
x y z=1
\end{gathered}
$$
The second equation can also be written as:

$$
x y+y z+z x=\frac{7}{2}
$$

Thus, $x, y, z$ are the roots of the following equation:

$$
v^{3}-\frac{7}{2} v^{2}+\frac{7}{2} v-1=0
$$

from which

$$
\begin{array}{r}
v_{1}=x_{1}=y_{2}=z_{3}=1 \\
v_{2}=x_{2}=y_{3}=z_{1}=2 \\
v_{3}=x_{3}=y_{1}=z_{2}=\frac{1}{2}
\end{array}
$$

(István Szabó.)

Number of solutions: 52.