如附件,謝謝!

設  
\( \vec{AN} \)  = x \( \vec{AC} \)
已知
\( \vec{AM} \)  = \(\frac{2}{3}\) \( \vec{AB} \)

\( \vec{AP} \)
= \(\frac{1}{3}\) \( \vec{AB} \) +  \(\frac{2}{5}\)  \( \vec{AC} \)
= \(\frac{1}{2}\) \( \vec{AM} \) +  \(\frac{2}{5x}\)  \( \vec{AN} \)
利用 M、P、N共線  \(\frac{1}{2}\) +  \(\frac{2}{5x}\) = 1
推得 x = \(\frac{1}{4}\)  
故 \( \vec{AN} \) : \( \vec{NC} \) = 4 : 1

第2題
\(\begin{align}
  & \overline{OQ}=a,\overline{OR}=b \\
& {{a}^{2}}+{{b}^{2}}=36 \\
& P\left( \frac{2}{3}a,\frac{1}{3}b \right) \\
& OAPB=\Delta OBP+\Delta OAP=a+\frac{1}{3}b \\
& {{\left( a+\frac{1}{3}b \right)}^{2}}\le \left( {{a}^{2}}+{{b}^{2}} \right)\left[ {{1}^{2}}+{{\left( \frac{1}{3} \right)}^{2}} \right]=40 \\
& OAPB\le 2\sqrt{10} \\
\end{align}\)

感謝thepiano老師的解說,我明白了!