Školjka

%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

%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 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 Let $a, b, c$ be positive integers satisfying the conditions $b > 2a$ and $c > 2b.$ Show that there exists a real number $\lambda$ with the property that all the three numbers $\lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $\left(\frac {1}{3}, \frac {2}{3} \right].$

%V0 Let $a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $k > 0,$ $$a^k_1 + a^k_2 + \ldots + a^k_n \geq 0.$$ Let $p = max\{|a_1|, \ldots, |a_n|\}.$ Prove that $p = a_1$ and that $$(x - a_1) \cdot (x - a_2) \cdots (x - a_n) \leq x^n - a^n_1$$ for all $x > a_1.$

%V0 Let $a$ and $b$ be non-negative integers such that $ab \geq c^2,$ where $c$ is an integer. Prove that there is a number $n$ and integers $x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that $$\sum^n_{i=1} x^2_i = a, \sum^n_{i=1} y^2_i = b, \text{ and } \sum^n_{i=1} x_iy_i = c.$$