+179 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.
+876 I think it is double dollar signs:
$$0$$
Hint: Multiply the equations by xyz.
+876 No problem! Also, if you want LaTeX text, you can use the \text{} environment:
$$\text{Wow, this is clearer text!}$$
+179 $$ \text{cool!} $$
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?
+179 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?
