RO-NMO 2011 Problem 104

a) Show that there exists exactly a sequence $\left( x_n,y_n \right)_{n\ge 0}$ of pairs of nonnegative integers, that satisfy the property that $\left( 1+\sqrt 33 \right)^n=x_n+y_n\sqrt 33,$ for all nonegative integers $n.$ b) Having in mind the sequence from a), prove that, for any natural prime $p,$ at least one of the numbers $y_{p-1} ,y_p$ and $y_{p+1}$ are divisible by $p.$