Školjka

%V0 Let $S$ be the set of all pairs $(m,n)$ of relatively prime positive integers $m,n$ with $n$ even and $m < n.$ For $s = (m,n) \in S$ write $n = 2^k \cdot n_o$ where $k, n_0$ are positive integers with $n_0$ odd and define $$f(s) = (n_0, m + n - n_0).$$ Prove that $f$ is a function from $S$ to $S$ and that for each $s = (m,n) \in S,$ there exists a positive integer $t \leq \frac{m+n+1}{4}$ such that $$f^t(s) = s,$$ where $$f^t(s) = \underbrace{ (f \circ f \circ \cdots \circ f) }_{t \text{ times}}(s).$$ If $m+n$ is a prime number which does not divide $2^k - 1$ for $k = 1,2, \ldots, m+n-2,$ prove that the smallest value $t$ which satisfies the above conditions is $\left [\frac{m+n+1}{4} \right ]$ where $\left[ x \right]$ denotes the greatest integer $\leq x.$

%V0 The function $F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $n \geq 0,$ (i) $F(4n) = F(2n) + F(n),$ (ii) $F(4n + 2) = F(4n) + 1,$ (iii) $F(2n + 1) = F(2n) + 1.$ Prove that for each positive integer $m,$ the number of integers $n$ with $0 \leq n < 2^m$ and $F(4n) = F(3n)$ is $F(2^{m + 1}).$

%V0 If $a, b, c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that $$\sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}.$$

%V0 The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality $$\displaystyle f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}$$ a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$. b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution. c) Find all $a \in \mathbb{N}$ such that the equation $f(x)=ax$ has a unique solution.

%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 $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f\!\left(a\right)-f\!\left(b\right)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f\!\left(c\right)$ for some positive integer $c$. Proposed by Juhan Aru, Estonia