SE-SKOLORNAS 1977 Problem 6

Show that there are positive reals $a$ , $b$ , $c$ such that $$\left\{ \begin{array}{l} a^2 + b^2 + c^2 > 2 a^3 + b^3 + c^3 <2 a^4 + b^4 + c^4 > 2 \end{array} \right.$$