2011SP-I-11

對於 $s=2$ 及 $C=356$，可得

$\begin{array}{rcl}356& =& 2k_1+4k_2\\ 178& =& k_1+2k_2\dots \text{①}\end{array}$

對於 $s=5$ 及 $C=1250$，可得

$\begin{array}{rcl}1250& =& 5k_1+25k_2\\ 250& =& k_1+5k_2\dots \text{②}\end{array}$

$\text{②}–\text{①}$，可得

$\begin{array}{rcl}72& =& 3k_2\\ k_2& =& 24\dots \text{③}\end{array}$

把 $\text{③}$ 代入 $\text{①}$，可得

$\begin{array}{rcl}178& =& k_1+2(24)\\ k_1& =& 130\end{array}$

$\therefore C=130s+24s^2$。

對於 $s=6$，

$\begin{array}{rcl}C& =& 130(6)+24(6{)}^2\\ & =& 1644\end{array}$

$\begin{array}{rcl}130s+24s^2& =& 539\\ 24s^2+130s-539& =& 0\\ (4s-11)(6s+49)& =& 0\end{array}$

$\therefore s={\displaystyle \frac{11}{4}}$ 或 $s=-{\displaystyle \frac{49}{6}}$。

因為周界必為正數，所以其周界為 ${\displaystyle \frac{11}{4}}$。