Determine the smallest positive integer $q$ with the following property: for every integer $m$ with $1\leqslant m\leqslant 1006$, there exists an integer $n$ such that $$\dfrac{m}{1007}q<n<\dfrac{m+1}{1008}q$$.

Let $ABC$ be an acute triangle with circumcentre $O$. Let $\mathit{\Gamma}_B$ be the circle through $A$ and $B$ that is tangent to $AC$, and let $\mathit{\Gamma}_C$ be the circle through $A$ and $C$ that is tangent to $AB$. An arbitrary line through $A$ intersects $\mathit{\Gamma}_B$ again in $X$ and $\mathit{\Gamma}_C$ again in $Y$. Prove that $|OX|=|OY|$.

Does there exist a prime number whose decimal representation is of the form $3811\cdots11$ (that is, consisting of the digits $3$ and $8$ in that order, followed by one or more digits $1$)?

Let $n$ be a positive integer. For each partition of the set $\{1,2,\dots,3n\}$ into arithmetic progressions, we consider the sum $S$ of the respective common differences of these arithmetic progressions. What is the maximal value that $S$ can attain? (An arithmetic progression is a set of the form $\{a,a+d,\dots,a+kd\}$, where $a,d,k$ are positive integers, and $k\geqslant 2$; thus an arithmetic progression has at least three elements, and successive elements have difference $d$, called the common difference of the arithmetic progression.)