## IMO SL-7(2009)

Discussion on International Mathematical Olympiad (IMO)
Sazid Akhter Turzo
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### IMO SL-7(2009)

Find all functions f from the set of real numbers into the set of real
numbers which satisfy for all real x, y the identity
$f (x f (x+y)) = f (y f (x))+x^2$.
I'm trying to solve it for 5 hours but still couldn't solve ( ). I've prove its surjectivity but couldn't prove that it's injective.

nafistiham
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### Re: IMO SL-7(2009)

Sazid Akhter Turzo wrote:Find all functions f from the set of real numbers into the set of real
numbers which satisfy for all real x, y the identity
f (x f (x+y)) = f (y f (x))+x2.
I'm trying to solve it for 5 hours but still couldn't solve ( ). I've prove its surjectivity but couldn't prove that it's injective.
I think the first red part means multiplying $x,f(x+y)$ and the second one means $x^2$

[using latex would be good ]
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

*Mahi*
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### Re: IMO SL-7(2009)

Edited and latexed.
@Sazid: Why don't you use latex?
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

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### Re: IMO SL-7(2009)

plugging x=0 we get f(o)=f(yf(o)).this implies the functional equation is constant.but how?this is an sl prblm

*Mahi*
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### Re: IMO SL-7(2009)

zadid xcalibured wrote:plugging x=0 we get f(o)=f(yf(o)).this implies the functional equation is constant.but how?this is an sl prblm
What if $f(0)=0$?Then all we get is $f(0)=0$.
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi