IMO SL-7(2009)

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Sazid Akhter Turzo
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IMO SL-7(2009)

Unread post by Sazid Akhter Turzo » Fri Mar 09, 2012 10:22 pm

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 ( :cry: :( ). I've prove its surjectivity but couldn't prove that it's injective.

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nafistiham
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Re: IMO SL-7(2009)

Unread post by nafistiham » Sat Mar 10, 2012 12:03 am

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 ( :cry: :( ). 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 $x^2$

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

Unread post by *Mahi* » Sat Mar 10, 2012 12:33 am

Edited and latexed.
@Sazid: Why don't you use latex?
http://www.matholympiad.org.bd/forum/vi ... p?f=25&t=2
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zadid xcalibured
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Re: IMO SL-7(2009)

Unread post by zadid xcalibured » Sat Mar 10, 2012 2:13 am

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

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Re: IMO SL-7(2009)

Unread post by *Mahi* » Sat Mar 10, 2012 9:35 am

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 $f(0)=0$.
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zadid xcalibured
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Re: IMO SL-7(2009)

Unread post by zadid xcalibured » Sat Mar 10, 2012 10:56 am

oh sorry this only implies f(o)=o

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