6. Find all real solutions of the system of equations

$$
\left\{\begin{array}{l}
\frac{1}{x}=\frac{32}{y^{5}}+\frac{48}{y^{3}}+\frac{17}{y}-15 \\
\frac{1}{y}=\frac{32}{z^{5}}+\frac{48}{z^{3}}+\frac{17}{z}-15 \\
\frac{1}{z}=\frac{32}{x^{5}}+\frac{48}{x^{3}}+\frac{17}{x}-15
\end{array}\right.
$$

(A. B. Vladimirov)
Solution. Let $F(t)=32 t^{5}+48 t^{3}+17 t-15$. Then the system has the form $F\left(\frac{1}{y}\right)=\frac{1}{x}, F\left(\frac{1}{z}\right)=\frac{1}{y}$, $F\left(\frac{1}{x}\right)=\frac{1}{z}$. From this, it follows that $F\left(F\left(F\left(\frac{1}{x}\right)\right)\right)=\frac{1}{x}$. Note that $F(0.5)=0.5$ and the function $F(t)-t$ is strictly increasing. Therefore, if $t>0.5$, then $t < F(F(F(t)))$. But this means that $\frac{1}{x}=0.5$, and the original system has a unique solution $x=y=z=2$.

Criteria. 1 point is given for finding the solution. 3 points for the solution under the assumption $x=y=z$. If the uniqueness of the solution to $F(t)=t$ is not proven, 2 points are deducted.