Math Olympiads problem that I couldn't do.

[mtayab1994]

yea and for x<0:

f(0)=f(k^{2}+f(f(-k^{2})) which then equals

=f(-k^{2})-k^{2}

 

[micromass]

yea and for x<0:

f(0)=f(k^{2}f(f(-k^{2})) which then equals

Typo here??

In general, you are indeed correct that f(x)=-x+c are the only possible answers. But you still need to show that they indeed satisfy the equation!

 

[mtayab1994]

Typo here??

yea i fixed it before you posted and btw the proof is as follows:

f(x)=-x+cfor c+f(0)&lt;br /&gt; &lt;br /&gt; indeed: f(x^{2}+f(y))=x^{2}-f(y)+c&lt;br /&gt; &lt;br /&gt; as: f(y)=-y+c then f(x^{2}+f(y))=-x^{2}+y-c+c&lt;br /&gt; &lt;br /&gt; and finally f(x^{2}+f(y))=y-x^{2} for every x in ℝ

 

[micromass]

Seems all good!

 

[mtayab1994]

yes now I'm sure i have all 4 questions correct for the olympiads !

 

[mtayab1994]

Do you know anywhere i can find olympiad like problems beside the imo-official site?

 

[mtayab1994]

The people from the olympiads gave us another test and they kept this same problem from last week !