Question 76
Let $R$ denote the set of real numbers. Define the set $S = \{1, -1\}$ and the function $sign(x) : R \to S$ as
$$
sign(x) = \begin{cases} 1 & \text{if } x \ge 0; \\ -1 & \text{if } x \lt 0. \end{cases}
$$
Given an odd integer $n$, are there $n^2 + n$ real numbers $a_{ij}, b_i \in S$ ($1 \le i, j \le n$) such that for arbitrary $n$ numbers $x_1, \dots, x_n \in S$, the number $z$ computed by the following formulas
$$
y_i = \operatorname{sign}\left(\sum_{j=1}^{n} a_{ij}x_j\right), \quad \forall 1 \le i \le n;
$$
$$
z = \operatorname{sign}\left(\sum_{i=1}^{n} b_i y_i\right).
$$
always equals the product $x_1x_2\cdots x_n$?
$$
sign(x) = \begin{cases} 1 & \text{if } x \ge 0; \\ -1 & \text{if } x \lt 0. \end{cases}
$$
Given an odd integer $n$, are there $n^2 + n$ real numbers $a_{ij}, b_i \in S$ ($1 \le i, j \le n$) such that for arbitrary $n$ numbers $x_1, \dots, x_n \in S$, the number $z$ computed by the following formulas
$$
y_i = \operatorname{sign}\left(\sum_{j=1}^{n} a_{ij}x_j\right), \quad \forall 1 \le i \le n;
$$
$$
z = \operatorname{sign}\left(\sum_{i=1}^{n} b_i y_i\right).
$$
always equals the product $x_1x_2\cdots x_n$?