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Solve for $x$,$y$ and $z$ $\in $ $\mathbb{R}$ if

$$\begin{align} x^2+x-1=y \\

y^2+y-1=z\\

z^2+z-1=x \end{align}$$

My Try:

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- Prerequisites/Books for A First Course in Linear Algebra

if $x=y=z$ then the two triplets $(1,1,1)$ and $(-1,-1,-1)$ are the Solutions.

if $x \ne y \ne z$ Then we have

$$\begin{align} x(x+1)=y+1 \\

y(y+1)=z+1\\

z(z+1)=x+1 \end{align}$$

Multiplying all we get $$xyz=1 \tag{1}$$ and adding all we get

$$x^2+y^2+z^2=3 \tag{2}$$

Now from Original Equations

$$\begin{align} x^2=y+1-x\\

y^2=z+1-y\\

z^2=x+1-z \end{align}$$ Multiplying all and Using $(1)$ we get

$$(y+1-x)(z+1-y)(x+1-z)=1 $$ $\implies$

$$xy+yz+zx-3=(x-y)(y-z)(z-x) \tag{3}$$ I am unable to proceed further..

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using the RMS GM inequality

$$\sqrt{\frac{x^2+y^2+z^2}{3}}\geq\sqrt[3]{xyz}$$

with equality if and only if $x=y=z$, plugging in your values for $xyz$ and$x^2+y^2+z^2$ we get that

$$\sqrt{\frac{3}{3}}\geq\sqrt[3]{1}$$

$$1=1$$

thus the only possible solutons are those you already stated.

No solutions exist for real, distinct $(x,y,z)$.

Writing $f(x)=x^2+x-1$, the existence of such a solution would mean that $f$ has a real point of least period 3; that is, $f^3(x)=x$ for some real $x$ with $f(x)\neq1$ i.e. $x\neq \pm 1$. Sarkovsky’s theorem then implies that $f$ has points of arbitrary least period. In particular, there would exist $x\in\mathbb{R}-\{-1,1\}$ such that $f^2(x)=x$. But $$f^2(x) – x = (x^2+x-1)^2+(x^2+x-1)-1-x = x^4+2x^3-2x-1 = (x-1)(x+1)^3.$$ and so there are no points of least period 2. Hence $f$ has no real points of least period 3.

P.S.: I’m actually only using the simplest case of Sarkovsky’s theorem, namely that 3-cycles imply period 2-cycles. But the full theorem implies that the same conclusion follows for *every* least period other than one (that is, $f$ has fixed points but not cycles.) This means that we would be no no better off if we increased the number of variables and equations beyond 3: we would still find no nontrivial real solutions. (Though this can be proven much more simply by the same RMS-GM inequality stated by cirpis in his answer…)

$\textbf{Hint:}$Note that $\displaystyle x^2-x-1=y$ is equal $\displaystyle (x-\frac{1}{2})^2-\frac{5}{2}=y-\frac{1}{2}$, so if you substitute $x_1=x-\frac{1}{2}$, $y_1=y-\frac{1}{2}$ and $z_1=z-\frac{1}{2}$ you get:

$$x_1^2-\frac{5}{2}=y_1$$

$$y_1^2-\frac{5}{2}=z_1$$

$$z_1^2-\frac{5}{2}=x_1$$

Substitute (1) to (2), next (2) to (3), you get polynomial equation to calculate $x_1$. The same with $y_1$ and $z_1$.

$\textbf{1)}$ Substituting Eq. 1 into Eq. 2 gives $xy(x+1)=z+1$, and then substituting Eq. 2 into Eq. 3 gives $xyz(x+1)=x+1$; so $x=-1$ or $xyz=1$. By symmetry, we have that $y=-1$ or $xyz=1$ and $z=-1$ or $xyz=1$. Since $x=-1, y=-1, z=-1$ is a solution, all other solutions must satisfy $xyz=1$.

Since $x=1,y=1, z=1$ is clearly a solution, it is left to show that there are no other solutions satisfying $xyz=1$.

$\textbf{2)}$ If $x\ge1$, then $y=x(x+1)-1\ge2-1=1$ and $z=y(y+1)-1\ge2-1=1$; so $xyz=1\implies x=1, y=1, z=1$. Therefore we can assume $x<1$ and, by symmetry, $y<1$ and $z<1$.

$\textbf{3)}$ If $x, y, z >0$, then $0<x, y, z <1\implies xyz<1$; so two of the variables must be negative, and we can assume that $x<0, y<0, z>0$. Since $z<1$, $xy=\frac{1}{z}>1$. However, this is impossible since $xy=x(x^2+x-1)=x^3+x^2-x$, and $g(x)=x^3+x^2-x$ has a maximum for $x<0$ given by $g(-1)=1$

since $g^{\prime}(x)=(3x-1)(x+1)$.

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