MK-MO 1994 Problem 3

a) Let $x_1, x_2, ..., x_n$ ( $n> 2$ ) be negative real numbers and $x_1 + x_2 + ... + x_n = m.$ Determine the maximum value of the sum $S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n.$ b) Let $x_1, x_2, ..., x_n$ ( $n> 2$ ) be nonnegative natural numbers and $x_1 + x_2 + ... + x_n = m.$ Determine the maximum value of the sum $S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n.$