Question 19
Find all real triples $(x,y,z)$ satisfying
$$
\begin{array}{cccc}
x+y+z & = & 1 \\
x^{2} y+y^{2} z+z^{2} x & = & x y^{2}+y z^{2}+z x^{2} \\
x^{3}+y^{2}+z & = & y^{3}+z^{2}+x
\end{array}
$$
$$
\begin{array}{cccc}
x+y+z & = & 1 \\
x^{2} y+y^{2} z+z^{2} x & = & x y^{2}+y z^{2}+z x^{2} \\
x^{3}+y^{2}+z & = & y^{3}+z^{2}+x
\end{array}
$$