[Symmetric Systems. Involutory Transformations]

Find all real solutions of the system of equations

$$
\begin{aligned}
& x^{2}+y^{2}+z^{2}=1, \\
& x^{3}+y^{3}+z^{3}=1 .
\end{aligned}
$$
For $0<|x|<1$, the inequality $x^{3}<x^{2}$ is satisfied.

## Solution

If $x \leq 1$, then $x^{3} \leq x^{2}$, and equality is achieved only when $x=0$ and $x=1$. From the equation $x^{2}+y^{2}+z^{2}=1$, it follows that $x, y, z$ are numbers between -1 and 1. Therefore, $x^{3}+y^{3}+z^{3} \leq x^{2}+y^{2}+z^{2}$, and equality is achieved when each of the numbers $x, y, z$ is either 0 or 1.

## Answer

$\{1,0,0\}$.