Školjka

%V0 Find all pairs of functions $f, g : \mathbb R \to \mathbb R$ such that $f \left( x + g(y) \right) = x \cdot f(y) - y \cdot f(x) + g(x)$ for all real $x, y$.

%V0 Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying $$f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)$$ for all $x,y$.

%V0 Does there exist a function $s\colon \mathbf{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$, then ${s(x)s(y)=-1}$? Justify your answer.

%V0 Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f\left(x+y\right)+f\left(x\right)f\left(y\right)=f\left(xy\right)+2xy+1$ for all real numbers $x$ and $y$.

%V0 Consider those functions $f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition $$f(m + n) \geq f(m) + f(f(n)) - 1$$ for all $m,n \in \mathbb{N}.$ Find all possible values of $f(2007).$ Author: unknown author, Bulgaria

%V0 Let $n$ be a positive integer, and let $x$ and $y$ be a positive real number such that $x^n + y^n = 1.$ Prove that $$ \left(\sum^n_{k = 1} \frac {1 + x^{2k}}{1 + x^{4k}} \right) \cdot \left( \sum^n_{k = 1} \frac {1 + y^{2k}}{1 + y^{4k}} \right) < \frac{1}{(1 - x)(1 - y)} \text{.}$$ Author: unknown author, Estonia