An ordered pair of integers $(m,n)$ with $1<m<n$ is said to be a Benelux couple if the following two conditions hold: $m$ has the same prime divisors as $n$, and $m+1$ has the same prime divisors as $n+1$. (a) Find three Benelux couples $(m,n)$ with $m\leqslant 14$. (b) Prove that there are infinitely many Benelux couples

Let $ABC$ be a triangle with incentre $I$. The angle bisectors $AI$, $BI$ and $CI$ meet $[BC]$, $[CA]$ and $[AB]$ at $D$, $E$ and $F$, respectively. The perpendicular bisector of $[AD]$ intersects the lines $BI$ and $CI$ at $M$ and $N$, respectively. Show that $A$, $I$, $M$ and $N$ lie on a circle.

If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded: $a_0 = p$ and $a_{n+1} = 3a_n-2\mathrm{c}(a_n)$ for $n \geqslant 0$.

Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n + 1$ or $2n$, provided that the number is not greater than $N$. The player who writes $N$ on the blackboard wins. (a) Determine which player has a winning strategy if $N = 2011$. (b) Find the number of positive integers $N\leqslant2011$ for which Brian has a winning strategy. (This is based on ISL 2004, Problem C5.)