Školjka

%V0 Dokažite da ne postoji funkcija $f : \mathbb{R} \rightarrow \mathbb{R}$ koja zadovoljava ove uvjete: $(i)$ $f(1 + f(x)) = 1 - x$, za svaki $x \in \mathbb{R}$, $(ii)$ $f(f(x)) = x$, za svaki $x \in \mathbb{R}$.

%V0 Funkcija $f$ definirana je za svaki realni broj i za svako $x$ i $y$ vrijedi $$ f(xy)=f(x)\cdot f(y)-f(x+y)+1, $$ pri čemu je $f(1)=2$. Odredite $f(m)$ za svaki cijeli broj $m$.

%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 $x,y$.

%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 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 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