What are all possible functions on the reals that satisfy this functional equation?

$$f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$$

source: http://www.imo2015.org/solution.php?lang=en

Question

What are all possible functions on the reals that satisfy this functional equation?

$$f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$$

source: http://www.imo2015.org/solution.php?lang=en

jtvatsim · Answer

Well... the identity function works for starters... :)

empty · Answer

Haha yeah, the trouble is showing this is the only possible function.

danica518 · Answer

really thats the only one?

jtvatsim · Answer

In fact, f(x) = c does not appear to work for any constant....

empty · Answer

No, that's just wishful thinking coming in, that or find all the other solutions whatever they may be.

empty · Answer

I have no idea what the general solution is!

danica518 · Answer

can a function be both multiplicative and also obey lienar Super position
f(xy)=f(x)+f(y) and f(x)*F(y)?

danica518 · Answer

ok i guess not

anonymous · Answer

@danica518 the obvious solution $f(x)=x$ is both

danica518 · Answer

other than identity

danica518 · Answer

but the identity works? lemme see

danica518 · Answer

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