The question is from the Bangladesh Math Olympiad- Barisal division from $2017$.
$a$, $b$ and $c$ are three separate real numbers, where $a+\frac 1b=b+\frac 1c=c+\frac 1a$. What is the product of the absolute values of $a$, $b$ and $c$?
If they were not different, then $a=b=c \neq 0$ would do it but the real numbers are different so I am confused.
I know that none of $a,b$ and $c$ are equal to $0$.
I can do $a+\frac 1b=b+\frac 1a$ by letting $a=-\frac 1b$ but I can't extend this to $3$ variables. I am fairly sure that at least one of the numbers has to be negative, which might be why they specified absolute values. I have tried some random values but nothing works.
I would like to specify that this is not a homework problem, it is part of my preparation for an upcoming olympiad. This is the first problem of this sort that I have encountered, which is why I would prefer hints and general solutions over a complete solution. Thank you for your time.