\[\begin{array}{l}
{a^2} + {b^2} + {c^2} = {a^3} + {b^3} + {c^3} = 1\\
\to {a^2}(1 - a) + {b^2}(1 - b) + {c^2}(1 - c) = 0\\
{a^2} + {b^2} + {c^2} = 1 \to {a^2} \le 1 \to \left| a \right| \le 1 \to (1 - a) \ge 0,1 - b \ge 0;1 - c \ge 0\\
\to {a^2}(1 - a) + {b^2}(1 - b) + {c^2}(1 - c) \ge 0 \to (a,b,c) = (1,0,0);(0,1,0);(0,0,1)\\
\to {a^5} + {b^5} + {c^5} = 1 + 0 + 0 = 1
\end{array}\]
