5.
\((1)\)若\(x\),\(y\)满足\(\begin{cases} & 2x-y\leqslant 0 \\ & x+y\leqslant \\ & x\geqslant 0 \\ \end{cases}\),则\(2x+y\)的最大值为________.
\((2)\)若\(P=\dfrac{{{2017}^{2016}}+1}{{{2017}^{2017}}+1}\),\(Q=\dfrac{{{2017}^{2017}}+1}{{{2017}^{2018}}+1}\),则\(P\)和\(Q\)的大小关系为\(P\)________\(Q(\)用“\( > \)”、“\( < \)”或“\(=\)”填空\()\).
\((3)\)等差数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\),若\(a_{2}+a_{4}+a_{9}=24\),则\(\dfrac{{{S}_{8}}}{8}\cdot \dfrac{{{S}_{10}}}{10}\)的最大值为________.
\((4)\)在等边三角形\(ABC\)中,\(P\)为三角形\(ABC\)内一动点,且\(∠BPC=120^{\circ}\),则\(\dfrac{PA}{PC}\)的最小值为________.