IMO Shortlist 2009 A7
- asif e elahi
- Posts:185
- Joined:Mon Aug 05, 2013 12:36 pm
- Location:Sylhet,Bangladesh
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
Re: IMO Shortlist 2009 A7
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
-
- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Re: IMO Shortlist 2009 A7
Elaborate the surjectivity part, please.
Re: IMO Shortlist 2009 A7
We have \(f(xf(x))=x^2\) and \(f(-xf(x))=-x^2\). Shift \(x\to \sqrt x\) to get \(f\left(\sqrt xf\left(\sqrt x\right)\right)=x\) and \(f\left(-\sqrt x f\left(\sqrt x\right)\right)=-x\), true for all \(x\in \mathbb{R}^+\). Also we have \(f(0)=0\). So for all \(x\in\mathbb{R}\), we have some \(n\in\mathbb{R}\) (namely \(n=\pm\sqrt x f\left(\sqrt x\right)\)) so that \(f(n)=x\), proving surjectivity.
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.