Školjka

%V0 Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1,2,\cdots \ 2p\}$ are there, the sum of whose elements is divisible by $p$?

%V0 Does there exist an integer $n > 1$ which satisfies the following condition? The set of positive integers can be partitioned into $n$ nonempty subsets, such that an arbitrary sum of $n - 1$ integers, one taken from each of any $n - 1$ of the subsets, lies in the remaining subset.

%V0 Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.

%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