Školjka

%V0 Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$.

%V0 Determine that all $k \in \mathbb{Z}$ such that $\forall n$ the numbers $4n+1$ and $kn+1$ have no common divisor.

%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 $n$ 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