RMM 2010 Problem 2

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

Marko Radovanović, Serbia