2015 IMO Problems/Problem 5

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f$ : $\mathbb{R}\rightarrow\mathbb{R}$ satisfying the equation

$f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)$

for all real numbers $x$ and $y$ .

Proposed by Dorlir Ahmeti, Albania

This problem needs a solution. If you have a solution for it, please help us out by adding it. Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f$ : $\mathbb{R}\rightarrow\mathbb{R}$ satisfying the equation

$f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)$

for all real numbers $x$ and $y$ .

Proposed by Dorlir Ahmeti, Albania

This problem needs a solution. If you have a solution for it, please help us out by adding it.

All forms of a function can be expressed in a major form as as $f(r) = ar^t + c$ using this, $f(x+f(x+y))+f(xy) = a(x + a(x+y)^t + c)^t + c +a(xy)^t + c$ and $x+f(x+y)+yf(x) =x + a(x+y)^t + c + ay(x)^t +yc$ and for both expressions to be equal, $t$ has to be 1, $c$ has to be 0, and $a$ has to be 1. therefore the function that can satisfy the equation in the question is $f(r) = r^1 + 0$ which is $f(r) = r$ .

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2015 IMO Problems/Problem 5