Školjka

%V0 Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions: (1) the set of prime divisors of the elements in $A$ consists of $p-1$ elements; (2) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power. What is the largest possible number of elements in $A$ ?

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

%V0 For a prime $p$ and a given integer $n$ let $\nu_p(n)$ denote the exponent of $p$ in the prime factorisation of $n!$. Given $d \in \mathbb{N}$ and $\{p_1,p_2,\ldots,p_k\}$ a set of $k$ primes, show that there are infinitely many positive integers $n$ such that $d|\nu_{p_i}(n)$ for all $1 \leq i \leq k$. Author: Tejaswi Navilarekkallu, India

%V0 Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $\left(a^n-1\right)\left(b^n-1\right)$ is not a perfect square. Proposed by Mongolia