2013 Putnam A6

Define a function $w: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$
as follows. For $\left| a \right|, \left| b \right| \leq 2$,
let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b) = 0$.

\begin{tabular}{|cc|r|r|r|r|r|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{$w(a,b)$}} & \multicolumn{5}{|c|}{$b$} \\
& & -2 & -1 & 0 & 1 & 2 \\
\hline
& -2 & -1 & -2 & 2 & -2 & -1 \\
& -1 & -2 & 4 & -4 & 4 & -2 \\
$a$ & 0 & 2 & -4 & 12 & -4 & 2 \\
& 1 & -2 & 4 & -4 & 4 & -2 \\
& 2 & -1 & -2 & 2 & -2 & -1 \\
\hline
\end{tabular}

For every finite subset $S$ of $\mathbb{Z} \times \mathbb{Z}$,

define

$$

A(S) = \sum_{(\mathbf{s}, \mathbf{s}') \in S \times S} w(\mathbf{s} - \mathbf{s}').

$$

Prove that if $S$ is any finite nonempty subset of $\mathbb{Z} \times \mathbb{Z}$, then $A(S) > 0$.

(For example, if $S = \{(0,1), (0,2), (2,0), (3,1)\}$, then the terms in $A(S)$ are $12, 12, 12, 12, 4, 4, 0, 0, 0,0,-1,-1,-2,-2,-4,-4$.)

定义函数 $w: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$

如下。对于\left|一个\右|，\左| b \右| \leq 2,

设 $w(a,b)$ 如表所示；否则，令 $w(a,b) = 0$。

\begin{表格}{|cc|r|r|r|r|r|}

\h行

\multicolumn{2}{|c|}{\multirow{2}{*}{$w(a,b)$}} & \multicolumn{5}{|c|}{$b$} \\

&& -2 & -1 & 0 & 1 & 2 \\

\h行

& -2 & -1 & -2 & 2 & -2 & -1 \\

& -1 & -2 & 4 & -4 & 4 & -2 \\

$a$ & 0 & 2 & -4 & 12 & -4 & 2 \\

& 1 & -2 & 4 & -4 & 4 & -2 \\

& 2 & -1 & -2 & 2 & -2 & -1 \\

\h行

\end{表格}

对于 $\mathbb{Z} \times \mathbb{Z}$ 的每个有限子集 $S$，

定义

$$

A(S) = \sum_{(\mathbf{s}, \mathbf{s}') \in S \times S} w(\mathbf{s} - \mathbf{s}')。

$$

证明如果 $S$ 是 $\mathbb{Z} \times \mathbb{Z}$ 的任意有限非空子集，则 $A(S) > 0$。

(例如，如果 $S = \{(0,1), (0,2), (2,0), (3,1)\}$，则 $A(S)$ 中的项为 $12, 12, 12, 12, 4, 4, 0, 0, 0,0,-1,-1,-2,-2,-4,-4$。)