MHB An intriguing system of equation

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The discussion revolves around solving a system of equations involving three variables, x, y, and z, in terms of a parameter k. The initial solutions found are when x, y, and z are equal, specifically at values of ±√7a and 0. The user struggles to prove that the system only has these three real roots, despite attempts using various mathematical techniques. A key insight is the cyclic nature of the equations, suggesting that if one variable is less than another, it leads to contradictions unless all variables are equal. The conclusion reached is that the only valid solution occurs when x equals y equals z.
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Hi MHB,

I've recently come across a system of equation problem that I could not solve it fully, I don't know how to prove that system, the resulted polynomial has only three real roots.

Problem:

Solve for all real solutions in terms of $k$ for the system below:

$x(x^2+9k^2)=2z(x^2+k^2)$

$y(y^2+9k^2)=2x(y^2+k^2)$

$z(z^2+9k^2)=2y(z^2+k^2)$

It's no hard to see that when $x=y=z$, then we have the solutions:

$(x,\,y,\,z)=(-\sqrt{7}a,\,-\sqrt{7}a,\,-\sqrt{7}a)\stackrel{\text{or}}{=}(0,\,0,\,0)\stackrel{\text{or}}{=}(\sqrt{7}a,\,\sqrt{7}a,\,\sqrt{7}a)$

But I don't know what other conclusions that I could draw when I considered the condition where $x\ne y \ne z$ that could lead me to solve for this problem.

Wolfram has confirmed that those are the only real solutions the the given system, and I tried, using the knowledge that I borrowed from inequality, geometry, function, etc and nope, nothing helped. And here I am, hoping to gain some useful advice from the members to solve this system successfully.

Any help would be much appreciated.

Thanks.
 
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This is just a hunch but the cyclic nature of the system along with the only solution being $x = y = z$ suggests that $x < y$ implies $y < x$ and vice versa, and same for $(y, z)$ and $(z, x)$ (or the other way around) so that the only solution can be $x = y = z$. I haven't tried, but if you haven't tried that it seems like it may be fruitful.
 
Thanks Bacterius for your reply and yes, I think I was able to complete the solution based on your suggestion and here is my work:

First, if we rewrite the first given equation as $\dfrac{2z}{x}=\dfrac{x^2+9k^2}{x^2+k^2}$, we see that $\dfrac{2z}{x}>0$ for all real $x$ and $k$.

This means both $x$ and $z$ are either both positive or both negative. By the same token, we also have $x$ and $y$ are both positive or negative, and so are $y$ and $z$.

Since $x^2,\,9k^2,\,k^2$ are all positive and if we apply AM-GM inequality on those, we see that

$x^2+9k^2\ge 2(\sqrt{9x^2k^2})=2(3xk)=6xk$ and $x^2+k^2\ge 2(\sqrt{x^2k^2})=2(xk)=2xk$

Now, if we consider the case where $x$ and $z$ are both positive, we see that

$\dfrac{2z}{x}=\dfrac{x^2+9k^2}{x^2+k^2}\ge \dfrac{6xk}{2xk}=3$

$2z\ge 3x$

$\therefore z>x$

Similarly, from $\dfrac{2x}{y}=\dfrac{y^2+9k^2}{y^2+k^2}$, we have $x>y$ and this suggests $z>y$ but from $\dfrac{2y}{z}=\dfrac{z^2+9k^2}{z^2+k^2}$, we have $y>z$. This leads to a contradiction.

The other case for which all $x,\,y,\,z$ are negative works the same way and so, there is only one condition in the given system of equation that we need to consider, and that is when $x=y=z$.
 
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