Proof Writing & Problem Solving · Challenge problem

A draft proves an inequality about an expression $E(a, b, c)$ in positive real numbers and begins "Without loss of generality $a \le b \le c$." This is justified when $E$ is symmetric (unchanged by EVERY permutation of $a, b, c$). For how many of the following expressions is it justified?

$a + b + c$;  $a^2 b + b^2 c + c^2 a$;  $abc$;  $a^3 + b^3 + c^3 - 3abc$;

$|a - b| + |b - c| + |c - a|$;  $\dfrac{a}{b + c} + \dfrac{b}{c + a} + \dfrac{c}{a + b}$;  $\dfrac{a^2}{b} + \dfrac{b^2}{c} + \dfrac{c^2}{a}$;  $(a - b)^2 (b - c)^2 (c - a)^2$.