BxMO 2015

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

$$\frac{m}{1007} q<n<\frac{m+1}{1008} q .$$

Let $A B C$ be an acute triangle with circumcentre $O$. Let $\Gamma_{B}$ be the circle through $A$ and $B$ that is tangent to $A C$, and let $\Gamma_{C}$ be the circle through $A$ and $C$ that is tangent to $A B$. An arbitrary line through $A$ intersects $\Gamma_{B}$ again in $X$ and $\Gamma_{C}$ again in $Y$. Prove that $|O X|=|O Y|$.

The following solutions are valid for the configurations appearing in the diagrams.

Does there exist a prime number whose decimal representation is of the form $3811 \cdots 11$ (that is, consisting of the digits 3 and 8 in that order followed by one or more digits 1 )?

An arithmetic progression is a set of the form $\{a, a+d, \ldots, a+k d\}$, where $a, d, k$ are positive integers and $k \geqslant 2$. Thus an arithmetic progression has at least three elements and the successive elements have difference $d$, called the common difference of the arithmetic progression.

Let $n$ be a positive integer. For each partition of the set $\{1,2, \ldots, 3 n\}$ into arithmetic progressions, we consider the sum $S$ of the respective common differences of these arithmetic progressions. What is the maximal value $S$ that can attain? (A partition of a set $A$ is a collection of disjoint subsets of $A$ whose union is $A$.)