While I was working on a problem, which was

\( a+(1/b) = 7/3, b+(1/c) = 4, c + (1/a) = 1\). What is the value of abc

I wanted to generalize this question, i.e. I wanted to fetch out the important properties of these equations

I made some manipulations to the problem and noticed the property that if \(a+(1/b) = l, b+(1/c) = m, c+(1/a) = n,\) and \(abc = 1\) then
\(l = (2+m+n)/(mn-1)\)
\(m = (2+l+n)/(ln-1)\)
\(n = (2+l+m)/(lm-1)\)
and this gives,
\(lmn = (2+l+m)(2+l+n)(2+m+n)/[(lm-1)(ln-1)(mn-1)] = l+m+n+2\)

The interesting part about this property is how bad this expression looks and many good problems can be designed based on this property!
A simple problem on this which I thought of was:

If \(a+(1/b) = l, b+(1/c) = m, c+(1/a) = n,\) and \(abc = 1, l+m+n = 8\) then find the value of: \((2+l+m)(2+l+n)(2+m+n)/[(lm-1)(ln-1)(mn-1)]\)

I gave this problem to my friend, he later told me that while solving this problem his main goal was that he was trying to look for a way to cancel all the terms by substituting, and transposing the terms. But one can simply solve this problem by simple manipulations to the equations given!!