Školjka

%V0 Let $\mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $f: \mathbb{N} \mapsto \mathbb{N}$ satisfying $$f(m + f(n)) = n + f(m + 95)$$ for all $m$ and $n$ in $\mathbb{N}.$ What is the value of $\sum^{19}_{k = 1} f(k)?$

%V0 Show that there exists a bijective function $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $m,n\in \mathbb{N}_{0}$: $$f(3mn + m + n) = 4f(m)f(n) + f(m) + f(n).$$

%V0 Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1,\ldots b_n$ and $c_1,\ldots,c_n$ such that - for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$, and - the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$ Prove that $${1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.$$

%V0 Let $n$ be a positive integer, and let $x$ and $y$ be a positive real number such that $x^n + y^n = 1.$ Prove that $$ \left(\sum^n_{k = 1} \frac {1 + x^{2k}}{1 + x^{4k}} \right) \cdot \left( \sum^n_{k = 1} \frac {1 + y^{2k}}{1 + y^{4k}} \right) < \frac{1}{(1 - x)(1 - y)} \text{.}$$ Author: unknown author, Estonia

%V0 For every $n\in\mathbb{N}$ let $d(n)$ denote the number of (positive) divisors of $n$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ with the following properties: $d\left(f(x)\right) = x$ for all $x\in\mathbb{N}$. $f(xy)$ divides $(x - 1)y^{xy - 1}f(x)$ for all $x$, $y\in\mathbb{N}$. Proposed by Bruno Le Floch, France

%V0 Let $P\!\left(x\right)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n\!\left(x\right) = x$ is equal to $P\!\left(n\right)$ for every $n \geqslant 1$, where $T^n$ denotes the $n$-fold application of $T$. Proposed by Jozsef Pelikan, Hungary