Školjka

%V0 Does there exist a sequence $F(1), F(2), F(3), \ldots$ of non-negative integers that simultaneously satisfies the following three conditions? (a) Each of the integers $0, 1, 2, \ldots$ occurs in the sequence. (b) Each positive integer occurs in the sequence infinitely often. (c) For any $n \geq 2,$ $$F(F(n^{163})) = F(F(n)) + F(F(361)).$$

%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 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 Let $T$ denote the set of all ordered triples $\left(p,q,r\right)$ of nonnegative integers. Find all functions $f: T \rightarrow \mathbb{R}$ satisfying $$f(p,q,r) = \begin{cases} 0 &\text{if}\; pqr = 0,\\ 1+\frac{1}{6}(f(p+1,q-1,r)+f(p-1,q+1,r) &\\ +f(p-1,q,r+1)+f(p+1,q,r-1) &\\ +f(p,q+1,r-1)+f(p,q-1,r+1)) &\text{otherwise}\end{cases}$$ for all nonnegative integers $p$, $q$, $r$.

%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 Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form $$x^n+c_{n-1}x^{n-1}+...+c_1x+c_0$$ where each of $c_0,c_1,...,c_{n-1}$ is equal to $1$ or $-1$.