Školjka

%V0 Find all functions $f: \mathbb{R}^{ + }\to\mathbb{R}^{ + }$ satisfying $f\left(x + f\left(y\right)\right) = f\left(x + y\right) + f\left(y\right)$ for all pairs of positive reals $x$ and $y$. Here, $\mathbb{R}^{ + }$ denotes the set of all positive reals. Proposed by Paisan Nakmahachalasint, Thailand

%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 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 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 nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that (i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$.

%V0 Find all functions $f$ from the reals to the reals such that $$f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)$$ for all real $x,y$.