Školjka

%V0 Real numbers $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$ are given. For each $i$, $(1 \leq i \leq n )$, define $$d_{i} = \max \{ a_{j}\mid 1 \leq j \leq i \} - \min \{ a_{j}\mid i \leq j \leq n \}$$ and let $d = \max \{d_{i}\mid 1 \leq i \leq n \}$. (a) Prove that, for any real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, $$\max \{ |x_{i} - a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)$$ (b) Show that there are real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). Author: Michael Albert, New Zealand

%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 $-1$.

%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 $T$ 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 Let $a, b, c$ be positive real numbers so that $abc = 1$. Prove that $$\left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.$$

%V0 Let $a$, $b$, $c$ be positive real numbers such that $abc = 1$. Prove that $$\frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.$$