Question 183
Given a positive integer $n$ and $n^3$ integers $a_{ijk} \in \{1, -1\}$ ($1 \le i, j, k \le n$). Prove that there exist
$$
x_1, \dots, x_n, y_1, \dots, y_n, z_1, \dots, z_n \in \{1, -1\},
$$
such that the following inequality holds
$$
\left| \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} a_{ijk} x_i y_j z_k \right| \gt \frac{n^2}{3}
$$
$$
x_1, \dots, x_n, y_1, \dots, y_n, z_1, \dots, z_n \in \{1, -1\},
$$
such that the following inequality holds
$$
\left| \sum_{i=1}^{n} \sum_{j=1}^{n} \sum_{k=1}^{n} a_{ijk} x_i y_j z_k \right| \gt \frac{n^2}{3}
$$