Example 2. Determine all real solutions of the system of equations
$$
\left\{\begin{array}{l}
x+y+z=3, \\
x^{2}+y^{2}+z^{2}=3, \\
x^{5}+y^{5}+z^{5}=3
\end{array}\right.
$$

(2nd United States of America Mathematical Olympiad)
Let the plane $\pi: x+y+z=3$, the sphere $O: x^{2}+y^{2}+z^{2}=3$, and $d=\frac{|-3|}{\sqrt{3}}=\sqrt{3}=R$. By Theorem 2 (2), the plane $\pi$ is tangent to the sphere $O$, and the coordinates of the point of tangency are the solutions to the system of equations $\left\{\begin{array}{l}x+y+z=3, \\ x^{2}+y^{2}+z^{2}=3\end{array}\right.$. The solution is $x=1, y=1, z=1$, which also satisfies (3).
$\therefore$ The original system of equations has a unique solution $x=1, y=1, z=1$.