VWO 1999

Let $[mn]$ be a diameter of the circle $C$ and $[AB]$ a chord with given length on this circle. $[AB]$ neither coincides nor is perpendicular to $[MN]$ . Let $C,D$ be the orthogonal projections of $A$ and $B$ on $[MN]$ and $P$ the midpoint of $[AB]$ . Prove that $\angle CPD$ does not depend on the chord $[AB]$ .

Let $a,b,m,n$ integers greater than 1. If $a^n-1$ and $b^m+1$ are both primes, give as much info as possible on $a,b,m,n$ .