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Suppose that $f: \mathbb R \rightarrow \mathbb R$ such that

$$f(x^3+y^3)=f(x+y)((f(x-y))^2+f(xy)),$$ for all $x,y$ real numbers. Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? I did not come to any good result, but I think the solution should be difficult.

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    $\begingroup$ No, it is not true. There are two other constant solutions. $\endgroup$ – Andrés E. Caicedo Aug 26 '17 at 16:24
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    $\begingroup$ Let $f(x)=c$ be such that $c^2+c=1$. $\endgroup$ – Sándor Kovács Aug 26 '17 at 16:25
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No, the constant functions $$f=\frac{-1\pm\sqrt 5}{2}$$ are both solutions.

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    $\begingroup$ How do we show that these are the only solutions? $\endgroup$ – Noram Sadir Aug 26 '17 at 16:34
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    $\begingroup$ @NoramSadir: That would be the next question! $\endgroup$ – Stefan Kohl Aug 26 '17 at 16:36

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