Hint to puzzle 114: Sums of squares and cubes

Homogenize the inequality by multiplying the left-hand side by 1 = (abc)^1/3.  Then apply the Rearrangement Inequality (for three sequences), Muirhead's Inequality, or related tools.

Alternatively, show that, for R a positive real number and n a non-negative integer, Rⁿ(R − 1) (R − 1).