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 Find all surjective functions $f: \mathbb{N} \mapsto \mathbb{N}$ such that for every $m,n \in \mathbb{N}$ and every prime $p,$ the number $f(m + n)$ is divisible by $p$ if and only if $f(m) + f(n)$ is divisible by $p.$ Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran

%V0 For an integer $m$, denote by $t(m)$ the unique number in $\{1, 2, 3\}$ such that $m + t(m)$ is a multiple of $3$. A function $f: \mathbb{Z}\to\mathbb{Z}$ satisfies $f( - 1) = 0$, $f(0) = 1$, $f(1) = - 1$ and $f\left(2^{n} + m\right) = f\left(2^n - t(m)\right) - f(m)$ for all integers $m$, $n\ge 0$ with $2^n > m$. Prove that $f(3p)\ge 0$ holds for all integers $p\ge 0$. Proposed by Gerhard Woeginger, Austria

%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