VMO 1989 Problem 2

Let $a_1$ = 1, $a_2$ = 1, $a_{n+2}$ = $a_{n+1}$ + $a_n$ be the Fibonacci sequence. Show that there are infinitely many terms of the sequence such that $1985a_n^2$ + $1956a_n$ + 1960 is divisible by 1989. Does there exist a term such that $1985a_n^2$ + $1956a_n$ + 1960 + 2 is divisible by 1989?