第 2 題
參考 https://math.pro/db/viewthread.php?tid=2028

cefepime 兄這個解法實在是太高妙了

\(\begin{align}
  & {{x}^{2}}=3x+1 \\
& x-\frac{1}{x}=3 \\
& {{x}^{2}}+\frac{1}{{{x}^{2}}}=11 \\
& {{x}^{4}}+\frac{1}{{{x}^{4}}}=119 \\
&  \\
& \frac{{{x}^{10}}+{{x}^{8}}+{{x}^{2}}+1}{{{x}^{10}}+{{x}^{6}}+{{x}^{4}}+1} \\
& =\frac{\left( {{x}^{2}}+1 \right)\left( {{x}^{8}}+1 \right)}{\left( {{x}^{4}}+1 \right)\left( {{x}^{6}}+1 \right)} \\
& =\frac{{{x}^{8}}+1}{\left( {{x}^{4}}+1 \right)\left( {{x}^{4}}-{{x}^{2}}+1 \right)} \\
& =\frac{{{x}^{4}}+\frac{1}{{{x}^{4}}}}{\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)\left( {{x}^{2}}-1+\frac{1}{{{x}^{2}}} \right)} \\
& =\frac{119}{110} \\
\end{align}\)