Školjka

%V0 Suppose that $s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences $s_{s_1},s_{s_2},s_{s_3},\ldots$ and $s_{s_1 + 1},s_{s_2 + 1},s_{s_3 + 1},\ldots$ are both arithmetic progressions. Prove that the sequence $s_1,s_2,s_3, \ldots$ is itself an arithmetic progression. Proposed by Gabriel Carroll, USA

%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

%V0 Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations $$f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c).$$

%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 $x\to\infty$.