MN-MO 2007

Let $M$ be the midpoint of the side $BC$ of triangle $ABC$ . The bisector of the exterior angle of point $A$ intersects the side $BC$ in $D$ . Let the circumcircle of triangle $ADM$ intersect the lines $AB$ and $AC$ in $E$ and $F$ respectively. If the midpoint of $EF$ is $N$ , prove that $MN\parallel AD$ .

For all $n\ge2$ , let $a_n$ be the product of all coprime natural numbers less than $n$ . Prove that

(a) $n\mid a_n+1\Leftrightarrow n=2,4,p^\alpha,2p^\alpha$ (b) $n\mid a_n-1\Leftrightarrow n\ne2,4,p^\alpha,2p^\alpha$ Here $p$ is an odd prime number and $\alpha\in\mathbb N$ .