Školjka

%V0 Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.

%V0 Let ${\mathbb Q}^ +$ be the set of positive rational numbers. Construct a function $f : {\mathbb Q}^ + \rightarrow {\mathbb Q}^ +$ such that $$f(xf(y)) = \frac {f(x)}{y}$$ for all $x$, $y$ in ${\mathbb Q}^ +$.

%V0 Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

%V0 Let $S$ be the set of all real numbers strictly greater than −1. Find all functions $f: S \to S$ satisfying the two conditions: (a) $f(x + f(y) + xf(y)) = y + f(x) + yf(x)$ for all $x, y$ in $S$; (b) $\frac {f(x)}{x}$ is strictly increasing on each of the two intervals $- 1 < x < 0$ and $0 < x$.

%V0 Find all functions $f$ from the reals to the reals such that $$\left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz)$$ for all real $x,y,z,t$.

%V0 Let $n$ be a positive integer. Consider $$S = \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x + y + z > 0 \right \}$$ as a set of $(n + 1)^{3} - 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0,0,0)$. Author: Gerhard Wöginger, Netherlands