From time to time some user asks for help to prove a cyclic inequality, that is, something like $$f(x,y,z)\le k$$ where $x,y,z$ are usually real positive numbers and $f$ is a 'cyclic function' (I don't know the standard name for this), that is, it is a function that satisfies $$f(x,y,z)=f(z,x,y)$$ There is sometimes an additional condition of the from $g(x,y,z)=r$ where $g$ is also a cyclic function.
Example 1, example 2 (there are many more).
I have tried many of them, but I feel useless (I seldom manage to prove them, only the easiest ones), and, from the answers to these questions, I see that there is no 'magic' theorem or technique.
My question: is there any book with selected exercises, from easy to hard, to get trained in this kind of problems?