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 We denote by $\mathbb{R}^+$ the set of all positive real numbers. Find all functions $f: \mathbb R^ + \rightarrow\mathbb R^ +$ which have the property: $$f\left(x\right)f\left(y\right) = 2f\left(x + yf\left(x\right)\right)$$ for all positive real numbers $x$ and $y$.

%V0 Let $a_{ij}$ (with the indices $i$ and $j$ from the set $\left\{1,\ 2,\ 3\right\}$) be real numbers such that $a_{ij}>0$ for $i = j$; $a_{ij}<0$ for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers $a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3}$, $a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3}$, $a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}$ are either all negative, or all zero, or all positive.

%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 $x,y$.