Školjka

%V0 Does there exist a positive integer $n$ such that $n$ has exactly 2000 prime divisors and $n$ divides $2^n + 1$?

%V0 Let $b$ be an integer greater than $5$. For each positive integer $n$, consider the number $$x_n = \underbrace{11\cdots1}_{n - 1}\underbrace{22\cdots2}_{n}5,$$ written in base $b$. Prove that the following condition holds if and only if $b = 10$: there exists a positive integer $M$ such that for any integer $n$ greater than $M$, the number $x_n$ is a perfect square.

%V0 Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

%V0 Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$. Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.

%V0 Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

%V0 Prove that there are infinitely many positive integers $n$ such that $n^{2} + 1$ has a prime divisor greater than $2n + \sqrt {2n}$. Author: Kestutis Cesnavicius, Lithuania