S367. Solve in positive real numbers the system of equations: \begin{gather*} (x^3+y^3)(y^3+z^3)(z^3+x^3)=8,\\ \frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32. \end{gather*} Proposed by Nguyen Viet Hung, Hanoi University of Science, Vietnam
From https://www.awesomemath.org/wp-pdf-files/math-reflections/mr-2016-02/mr_2_2016_problems.pdf
I think I am smelling inequalities here. In the first equation I used Holder's inequality to show, $xyz \le 1$ , But in the second equation I used Titu's Lemma to get $x+y+z \le 3$ .But I think there would an equality case in one of the two equations. Can anyone help? The original source is Facebook 