#### Function IMO

###### back to index

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 $y$. Proposed by Dorlir Ahmeti, Albania

Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds: $f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).$ (Here $\mathbb{Z}$ denotes the set of integers.) Proposed by Liam Baker, South Africa

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$, $$f (f(x)f(y)) + f(x + y) = f(xy)$$