# Uniqueness of a solution of the system of equations

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 $(x, y, z)=(1, 1, 1)$. $$\begin{cases} x+y^2+z^3=3\\ y+z^2+x^3=3\\ z+x^2+y^3=3 \end{cases}$$

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What have you tried so far? –  TZakrevskiy Feb 18 '14 at 14:43
What rihkddd mentioned. I tried to manipulate it into an equation of the form $(y−x)^2A+(z−y)^2B+(x−z)^2C=0$ where $A,B,C$ are perfect squares, to show symmetry. I got $(y−x)A+(z−y)B+(x−z)C=0$, but that is not exactly what I am looking for. –  Emre Feb 18 '14 at 19:15

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).
Yes, under the hypothesis that $x=y=z$ you can easily show that the only solution is $x=y=z=1$. However, the true question is whether there're other, non-symmetrical solutions. –  TZakrevskiy Feb 18 '14 at 15:00