## IMO SL-7(2009)

- Sazid Akhter Turzo
**Posts:**69**Joined:**Sat Feb 18, 2012 9:15 am**Location:**Sirajganj-
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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.

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
**Posts:**829**Joined:**Mon Oct 17, 2011 3:56 pm**Location:**24.758613,90.400161-
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### Re: IMO SL-7(2009)

I think the first red part means multiplying $x,f(x+y)$ and the second one means $x^2$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.

[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.

Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

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

Edited and latexed.

@Sazid: Why don't you use latex?

http://www.matholympiad.org.bd/forum/vi ... p?f=25&t=2

@Sazid: Why don't you use latex?

http://www.matholympiad.org.bd/forum/vi ... p?f=25&t=2

Please read Forum Guide and Rules before you post.

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

Nur Muhammad Shafiullah | Mahi

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

Nur Muhammad Shafiullah | Mahi

- zadid xcalibured
**Posts:**217**Joined:**Thu Oct 27, 2011 11:04 am**Location:**mymensingh

### 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

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

What if $f(0)=0$?Then all we get is $f(0)=0$.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

Please read Forum Guide and Rules before you post.

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

Nur Muhammad Shafiullah | Mahi

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

Nur Muhammad Shafiullah | Mahi

- zadid xcalibured
**Posts:**217**Joined:**Thu Oct 27, 2011 11:04 am**Location:**mymensingh

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

oh sorry this only implies f(o)=o