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 $\mathbb{N} = \{1,2,3, \ldots\}$. 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 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 Prove that for all positive real numbers $a,b,c$, $$\frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1.$$

%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 $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that $$\frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 .$$ Hojoo Lee, Korea