Školjka

%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 We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating.

%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 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 $\frac{\tau (n^{2})}{\tau (n)}=m$.

%V0 For any positive integer $k$, let $f_k$ be the number of elements in the set $\{ k + 1, k + 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $m$, there exists at least one positive integer $k$ such that $f(k) = m$. (b) Determine all positive integers $m$ for which there exists exactly one $k$ with $f(k) = m$.