TITLE: Is it possible to have three real numbers that have both their sum and product equal to $1$?
QUESTION [59 upvotes]: I have to solve $ x+y+z=1$  and $xyz=1$ for a set of $(x, y, z)$. Are there any such real numbers? 
Edit : What if $x+y+z=xyz=r$, $r$ being an arbitrary real number. Will it still be possible to find real $x$, $y$, $z$?

REPLY [3 votes]: A negative fact after many positive solutions: you'll never find a solution with 3 positive numbers! In fact in that case AM-GM holds $$\frac{x+y+z}{3}\geq \sqrt[3]{xyz} $$ which would give $$\frac{1}{3} \geq 1$$