Školjka

%V0 Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1|a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

%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 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 $a_0$, $a_1$, $a_2$, $\ldots$ 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 Determine that all $k \in \mathbb{Z}$ such that $\forall n$ the numbers $4n+1$ and $kn+1$ have no common divisor.

%V0 Prove: If the sum of all positive divisors of $n \in \mathbb{Z}^{+}$ is a power of two, then the number/amount of the divisors is a power of two.