Question 185
Given a positive integer $n$, find all $n$-tuples of real numbers $(x_1, x_2, \dots, x_n)$ such that
$$
f(x_1, x_2, \dots, x_n) = \sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \left| k_1 x_1 + k_2 x_2 + \cdots + k_n x_n - 1 \right|
$$
attains its minimum.
$$
f(x_1, x_2, \dots, x_n) = \sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \left| k_1 x_1 + k_2 x_2 + \cdots + k_n x_n - 1 \right|
$$
attains its minimum.