Školjka

%V0 Find all functions $f: (0, \infty) \mapsto (0, \infty)$ (so $f$ is a function from the positive real numbers) such that $$\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}$$ for all positive real numbes $w,x,y,z,$ satisfying $wx = yz.$ Author: Hojoo Lee, South Korea

%V0 Find all the functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x-f(y))=f(f(y))+xf(y)+f(x)-1$$ for all $x,y \in \mathbb{R}$.

%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 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 ${\mathbb Q}^ +$ 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 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 $x\to\infty$.