Question 160
Determine the smallest possible real constant $C$ such that the inequality
$$
\left|x^{3}+y^{3}+z^{3}+1\right| \leqslant C\left|x^{5}+y^{5}+z^{5}+1\right|
$$
holds for all real numbers $x, y, z$ satisfying $x+y+z=-1$.
$$
\left|x^{3}+y^{3}+z^{3}+1\right| \leqslant C\left|x^{5}+y^{5}+z^{5}+1\right|
$$
holds for all real numbers $x, y, z$ satisfying $x+y+z=-1$.