Školjka

%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

%V0 Let $k$ be a positive integer. Prove that the number $(4 \cdot k^2 - 1)^2$ has a positive divisor of the form $8kn - 1$ if and only if $k$ is even. Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above. Author: Kevin Buzzard and Edward Crane, United Kingdom

%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 Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that $$\frac{a^m+a-1}{a^n+a^2-1}$$ is itself an integer. Laurentiu Panaitopol, Romania

%V0 Let $a > b > c > d$ be positive integers and suppose that $$ac + bd = (b+d+a-c)(b+d-a+c).$$ Prove that $ab + cd$ is not prime.

%V0 For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.