IMO 2015 - Problem 5

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IMO 2015 - Problem 5

Unread post by Nirjhor » Wed Jul 15, 2015 1:51 am

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.

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Thanic Nur Samin
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Re: IMO 2015 - Problem 5

Unread post by Thanic Nur Samin » Sun Apr 02, 2017 9:50 pm

The solutions are $2-x$ and $x$. They satisfy the FE.

Let $P(x,y)$ denote the FE.

$$P(0,0)\Rightarrow f(f(0))=0$$
$$P(0,f(0))\Rightarrow f(0)=0,2$$.

Case $1$: $f(0)=2$

$$P(0,x)\Rightarrow f(f(x))-f(x)=2(x-1)$$

Which implies $f(x)=x$ is only possible when $x=1$.

$$P(x,1)\Rightarrow f(x+f(x+1))=x+f(x+1)$$

So $x+f(x+1)=1$ and so $f(x)=2-x$ for all $x$.

Case $2$: $f(0)=0$.

$$P(-1,1)\Rightarrow f(-1)=-1$$
$$P(1,-1)\Rightarrow f(1)=1$$
$$P(x,0)\Rightarrow f(x+f(x))=x+f(x)$$
$$P(0,x)\Rightarrow f(f(x))=f(x)$$
$$P(x-1,1)\Rightarrow f(x-1+f(x))=x-1+f(x)$$
$$P(1,x-1+f(x))\Rightarrow f(x+1+f(x))=x+1+f(x)$$
$$\Rightarrow f(x+f(x-1))=x+f(x-1)$$
$$P(x,-1)\Rightarrow f(x)=-f(-x)$$
$$P(x,-x)\Rightarrow f(x)-f(x^2)=x-xf(x)$$
$$P(-x,x)\Rightarrow -f(x)-f(x^2)=-x-xf(x)$$
Substracting the last equation from the second last,
$f(x)=x$ For all real $x$.
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