Školjka

%V0 Find all triplets of positive integers $(a,m,n)$ such that $a^m + 1 \mid (a + 1)^n$.

%V0 Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

%V0 Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.

%V0 Prove that the equation $\frac {x^{7} - 1}{x - 1} = y^{5} - 1$ doesn't have integer solutions!

%V0 Let $a > b > 1$ be relatively prime positive integers. Define the weight of an integer $c$, denoted by $w(c)$ to be the minimal possible value of $|x| + |y|$ taken over all pairs of integers $x$ and $y$ such that $ax + by = c$. An integer $c$ is called a local champion if $w(c) \geq w(c \pm a)$ and $w(c) \geq w(c \pm b)$. Find all local champions and determine their number.

%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