IMO-2011-A3

Discussion on International Mathematical Olympiad (IMO)
User avatar
Tahmid Hasan
Posts:665
Joined:Thu Dec 09, 2010 5:34 pm
Location:Khulna,Bangladesh.
IMO-2011-A3

Unread post by Tahmid Hasan » Tue Jul 24, 2012 8:18 pm

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy
$g(f(x+y))=f(x)+(2x+y)g(y)$
বড় ভালবাসি তোমায়,মা

User avatar
Tahmid Hasan
Posts:665
Joined:Thu Dec 09, 2010 5:34 pm
Location:Khulna,Bangladesh.

Re: IMO-2011-A3

Unread post by Tahmid Hasan » Tue Jul 24, 2012 8:22 pm

Let $P(x,y)$ be the assertion $g(f(x+y))=f(x)+(2x+y)g(y)$
$P(x,y)-P(y,x) \Rightarrow f(x)-f(y)=(2y+x)g(x)-(2x+y)g(y)$.
Let $Q(x,y)$ be the assertion $f(x)-f(y)=(2y+x)g(x)-(2x+y)g(y)$
$Q(1,0) \Rightarrow f(1)-f(0)=g(1)-2g(0)$.....(1)
$Q(x,0)-Q(x,1) \Rightarrow f(1)-f(0)=2xg(1)+g(1)-2xg(0)-2g(x)$.....(2)
From (1),(2) we get $g(x)=(g(1)-g(0))x+g(0)$
now we have two cases.
Case1 $\Rightarrow g(1)-g(0)=0$
Then $g(x)=g(0)=c \forall x \in \mathbb {R}$,where $c$ is a constant.
Plugging $g(x)=c$ in $P(x,y) \Rightarrow f(x)+2cx+cy=c \forall x,y \in \mathbb {R}$
which implies $c=0,f(x)=0$ whcih is indeed a solution.
Case2 $\Rightarrow g(1)-g(0) \neq 0$
We may write $g(x)=ax+b;a \in \mathbb {R}-\{0\},b \in \mathbb {R}$.
Plugging $g(x)=ax+b$ in $P(x,-x) \Rightarrow af(0)+b=f(x)+xg(-x)$
or,$f(x)=ax^2-bx+af(0)+b$,let $af(0)+b=c$
Then $f(x)=ax^2-bx+c$
Plugging these values in $P(0,x)$
$\Rightarrow a^2x^2-abx+ac+b=ax^2+bx+c \forall x \in \mathbb {R}$,Let it be $R(x)$.
$R(0) \Rightarrow ac+b=c$.....(3)
$R(1)-R(-1) \Rightarrow b(a-1)=0$
Again we have 2 cases.
Case2.1:$a=1$,so from (3) we get $b=0$
so $f(x)=x^2+c,g(x)=x$ which indeed satisfy $P(x,y)$.
Case2.2:$b=0$,so from (3) we get $c(a-1)=0$
If $a=1$ it becomes case 2.1
So $c=0$ implying $f(x)=ax^2,g(x)=ax$
Plugging these values in $P(x,0) \Rightarrow a=0$ or $a=1$
if $a=1;f(x)=x^2,g(x)=x$,which is a subset of solutions of Case2.1
else if $a=0;f(x)=0,g(x)=0$.
=============================================
Synthesis of solutions:
1.$f(x)=0,g(x)=0 \forall x \in \mathbb {R}$
2,$f(x)=x^2+c,g(x)=x \forall x \in \mathbb {R}$.
বড় ভালবাসি তোমায়,মা

Post Reply