Question 19
a) Find the largest possible value of the number
$$
x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n,
$$
if $x_1, x_2, \dots, x_n$ ($n \ge 2$) are non-negative integers and their sum is $2011$.
b) Find the numbers $x_1, x_2, \dots, x_n$ for which the maximum value determined at a) is obtained.
$$
x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n,
$$
if $x_1, x_2, \dots, x_n$ ($n \ge 2$) are non-negative integers and their sum is $2011$.
b) Find the numbers $x_1, x_2, \dots, x_n$ for which the maximum value determined at a) is obtained.