Školjka

%V0 Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

%V0 Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by $$x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots$$ has all of its terms relatively prime to $m$.

%V0 We define a sequence $\left(a_{1},a_{2},a_{3},...\right)$ by setting $$a_{n} = \frac {1}{n}\left(\left[\frac {n}{1}\right] + \left[\frac {n}{2}\right] + \cdots + \left[\frac {n}{n}\right]\right)$$ for every positive integer $n$. Hereby, for every real $x$, we denote by $\left[x\right]$ the integral part of $x$ (this is the greatest integer which is $\leq x$). a) Prove that there is an infinite number of positive integers $n$ such that $a_{n + 1} > a_{n}$. b) Prove that there is an infinite number of positive integers $n$ such that $a_{n + 1} < a_{n}$.

%V0 Let $a_0$, $a_1$, $a_2$, $\ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $\gcd (a_i, a_{i + 1}) > a_{i - 1}$. Prove that $a_n\ge 2^n$ for all $n\ge 0$. Proposed by Morteza Saghafian, Iran

%V0 A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P\!\left(x\right) = \left(x+a\right)\left(x+b\right)$. a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P\!\left(1\right)$, $P\!\left(2\right)$, ..., $P\!\left(50\right)$ are balanced. b) Prove that if $P\!\left(n\right)$ is balanced for all positive integers $n$, then $a=b$. Proposed by Jorge Tipe, Peru

%V0 Let $k$ be a positive integer. Show that if there exists a sequence $a_0$, $a_1$, ... of integers satisfying the condition $$a_n=\frac{a_{n-1}+n^k}{n} \text{,} \qquad \forall n \geqslant 1 \text{,}$$ then $k-2$ is divisible by $3$. Proposed by Turkey