Školjka

%V0 Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$, ..., $a_n$ satisfying $$a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1$$ for every $k$ with $2 \leqslant k \leqslant n-1$. Proposed by North Korea

%V0 Let $n$ be a positive integer. Show that the numbers $$\binom{2^n - 1}{0},\; \binom{2^n - 1}{1},\; \binom{2^n - 1}{2},\; \ldots,\; \binom{2^n - 1}{2^{n - 1} - 1}$$ are congruent modulo $2^n$ to $1$, $3$, $5$, $\ldots$, $2^n - 1$ in some order. Proposed by Duskan Dukic, Serbia

%V0 Let $a_1$, $a_2$, $\ldots$, $a_n$ be distinct positive integers, $n\ge 3$. Prove that there exist distinct indices $i$ and $j$ such that $a_i + a_j$ does not divide any of the numbers $3a_1$, $3a_2$, $\ldots$, $3a_n$. Proposed by Mohsen Jamaali, Iran

%V0 Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations $$a^n + pb = b^n + pc = c^n + pa$$ then $a = b = c$. Proposed by Angelo Di Pasquale, Australia

%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 $k$ 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 $m$.