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 $\mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $f$ from $\mathbb{N}_0$ to itself such that $$f(m + f(n)) = f(f(m)) + f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0.$$

%V0 Let $n \geq 2$ be a positive integer and $\lambda$ a positive real number. Initially there are $n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $A$ and $B$, with $A$ to the left of $B$, and letting the flea from $A$ jump over the flea from $B$ to the point $C$ so that $\frac {BC}{AB} = \lambda$. Determine all values of $\lambda$ such that, for any point $M$ on the line and for any initial position of the $n$ fleas, there exists a sequence of moves that will take them all to the position right of $M$.

%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