Find solutions of system where $\frac{xyz}{x+z}=\frac{mp}{m+p}$ and so on.

Question

Find solutions of $$\begin{cases} \frac{xyz}{x+z}=\frac{mp}{m+p} \\[1ex] \frac{xyz}{y+z}=\frac{np}{n+p} \\[1ex] \frac{xyz}{x+y}=\frac{mn}{m+n} \end{cases}, \\$$ where $mnp>0$

Through a bit of guesswork, I managed to find that $\left(\frac{\sqrt{mnp}}{m}, \frac{\sqrt{mnp}}{n}, \frac{\sqrt{mnp}}{p}\right)$ is a solution (as is the negative of those three), but I have no clue how to start proving these are the solotions with no other possible. Any clues are much appreciated!

Answer 1

Observations towards a solution. Fill in the gaps as needed.

• Show that $xyz \neq 0$.
• Write the equations as $ \frac{1}{xy} + \frac{1}{yz} = \frac{x+z}{xyz} = \frac{m+p}{mp} = \frac{1}{m} + \frac{1}{p}$ (resp).
• Hence show that $ \frac{1}{xy} = \frac{1}{p} \Rightarrow xy =p$ (resp).
• Hence show that $ y^2 = \frac{ (xy)(yz)}{xz} = \frac{pm}{n}$
• Hence $ y = \pm \frac{ \sqrt{mnp}}{n}$ (resp).
• Finally, check that the only solutions are $ \pm ( \frac{\sqrt{mnp}}{m},\frac{\sqrt{mnp}}{n},\frac{\sqrt{mnp}}{p})$ (EG to satisfy $xy = p$).