A friend asked me the following question several days ago, and we still do not have a solution.
Prove that the system of equations below has only the solution $(x, y, z)=(1, 1, 1)$. $$ \begin{cases} x+y^2+z^3=3\\ y+z^2+x^3=3\\ z+x^2+y^3=3 \end{cases} $$
On
The question should be wrong since at least it can be checked by Wolfram Alpha that it has more than one group of solution (in fact it has $27$ groups of solution).
maybe you could find that x,y,z are symmetyical ,which means x,y,z have no difference with each other ,so x=y=z.you can solve this original equation by transform to x+x^2+x^3=3,so x =1,the so the only solution is (x,y,z)=(1,1,1). I'm not native English spaker ,sorry for my bad English,hope you can understand what I mean.