I have tried more than an hour but couldn't solve it, can somebody please give me a clue?

$$f:\mathbb R\rightarrow\mathbb R$$

$$f(f(f(X)))+f(f(X))+X=3f(X).$$ Find $f(X)$

I know that $f(X)=X$ is a solution, and I know that the function being (obviously) injective might help, but that's all the useful stuff I could gather.

  • $\begingroup$ Where is this problem from? $\endgroup$
    – Sil
    Jul 27 '19 at 16:53

Probably not a complete solution, I tried $f(x)=kx$, $k$ constant, $k\ne0$, which results in $$(1 - 3 k + k^2 + k^3) x=0.$$ If valid for all $x$ then $$1 - 3 k + k^2 + k^3=0 \quad\Leftrightarrow\quad k=\{1,-1-\sqrt{2},-1+\sqrt{2}\}.$$


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