Question 44
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \dots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \dots, x_n$, such that $0 \le x_i \le 1$, satisfying
$$
|f_1(x_1) + f_2(x_2) + \dots + f_n(x_n) - x_1x_2\dots x_n| \ge C_n.
$$
$$
|f_1(x_1) + f_2(x_2) + \dots + f_n(x_n) - x_1x_2\dots x_n| \ge C_n.
$$