Find all the triples $(x,y,z)$ $\in$ $\Bbb R^+_0$ that verifies the following equation system

Question

Find all the triples $(x,y,z)$ $\in$ $\Bbb R^+_0$ that verifies the next equation system \begin{align*} x^2-y &= (z-1)^2\\ y^2-z &= (x-1)^2\\ z^2-x &= (y-1)^2 \end{align*} My try

Adding the 3 equations, expanding the RHS and then substracting terms of the LHS I determinated $x+y+z=3$, so $(1,1,1)$ is a trivial solution, but I don't know a way to find more solutions.

Any help?

Answer 1

You have $$y\le x^2\le z^4\le y^8$$

This implies $y\ge 1$.

You can do the same thing for $x$ and $z$.

Then, since $x+y+z=3$, $(1,1,1)$ is the only solution.

Answer 2

Any number squared is non-negative, so the right side of those equations are all non-negative.

There-fore, the left-hand sides also have to be non-negative:

$x^2 \ge y$

$y^2 \ge z$

$z^2 \ge x$

Since the numbers themselves are non-negative, on one hand, if $x \le 1$, then $1 \ge x \ge x^2 \ge y \ge y^2 \ge z \ge z^2 \ge x$, which means all three have to be equal to 1.

On the other hand, if $x > 1$, then $z^2 >1$, so $z > 1$, then $y^2 > 1$ and finally $y > 1$, so all three have to be greater than one. Of course, if they are all greater than one they can't sum up to 3, so they have to be equal to one.