MathDB
Esoteric Sign Calculation

Source: Taiwan 2014 TST3, Problem 1

July 18, 2014
functionlinear algebramodular arithmeticnumber theoryTaiwan

Problem Statement

Let R\mathbb R be the real numbers. Set S={1,−1}S = \{1, -1\} and define a function sign⁡:R→S\operatorname{sign} : \mathbb R \to S by sign⁡(x)={1if x≥0;−1if x<0. \operatorname{sign} (x) = \begin{cases} 1 & \text{if } x \ge 0; \\ -1 & \text{if } x < 0. \end{cases} Fix an odd integer nn. Determine whether one can find n2+nn^2+n real numbers aij,bi∈Sa_{ij}, b_i \in S (here 1≤i,j≤n1 \le i, j \le n) with the following property: Suppose we take any choice of x1,x2,…,xn∈Sx_1, x_2, \dots, x_n \in S and consider the values \begin{align*} y_i &= \operatorname{sign} \left( \sum_{j=1}^n a_{ij} x_j \right),   \forall 1 \le i \le n; \\ z &= \operatorname{sign} \left( \sum_{i=1}^n y_i b_i \right) \end{align*} Then z=x1x2…xnz=x_1 x_2 \dots x_n.
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