VMO 1986 Problem 1

$a_1$, $a_2$, ... , $a_n$ are real numbers such that 1/2 ≤ $a_i$ ≤ 5 for each i. The real numbers $x_1$, $x_2$, ... , $x_n$ satisfy $4x_i^2$ - $4a_ix_i$ + ($a_i$ - 1)$^2$ = 0. Let S = ($x_1$ + $x_2$ + ... + $x_n$)/n, $S_2$ = ($x_1^2$ + $x_2^2$ + ... + $x_n^2$)/n. Show that √$S_2$ ≤ S + 1.