Školjka

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

%V0 For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$.

%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