VMO 1990 Problem 1

-1 < a < 1. The sequence $x_1$, $x_2$, $x_3$, ... is defined by $x_1$ = a, $x_{n+1}$ = ( √(3 - $3x_n^2$) - $x_n$)/2. Find necessary and sufficient conditions on a for all members of the sequence to be positive. Show that the sequence is always periodic.