Let \(x, y,\) and \(z\) be nonzero real numbers, such that no two are equal, and

\(\begin{align} x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x} \end{align}\)

Find all possible numeric values of \(xyz \).

Thanks in advance! (also, as a bonus question, how do you center \(\LaTeX\) in web2.0calc? Thanks.

xCorrosive May 26, 2021

#1**0 **

I think it is double dollar signs:

$$0$$

Hint: Multiply the equations by xyz.

MathProblemSolver101 May 26, 2021

#3**0 **

No problem! Also, if you want LaTeX text, you can use the \text{} environment:

$$\text{Wow, this is clearer text!}$$

MathProblemSolver101
May 26, 2021

#4**0 **

$$ \text{cool!} $$

Anyway, I didn't exactly multiply by $xyz$, but now I'm at a point where I have this equation

$(y-x)(y-z)(z-x)= \frac{(y-x)(y-z)(z-x)}{x^2y^2z^2}$

Any pointers on how to continue from here?

xCorrosive
May 26, 2021

#5**0 **

Wait I have a thought.

There are only two cases in which these two equations are equal to each other.

The first case is when two the variables are $0$, effectively leading the equation to be zero.

The second case is when $x^2y^2z^2 = 1$.

We're looking for the square root of this, so does it mean that the only solutions are $-1$ and $1$. Or are there other possible solutions?

xCorrosive
May 26, 2021