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$\begin{array}{l}
1. \,\frac{{\sqrt 3 }}{{c{\rm{o}}{{\rm{s}}^2}x}} < 4\tan x \Leftrightarrow \sqrt 3 ({\tan ^2}x + 1) < 4\tan x\\
\Leftrightarrow \frac{1}{{\sqrt 3 }} < {\mathop{\rm t}\nolimits} {\rm{anx}} < \sqrt 3 \\
\Leftrightarrow \frac{\pi }{6} + k\pi < x < \frac{\pi }{3} + k\pi ,k \in Z
\end{array}$
$\left. \begin{array}{l}
2. \,\,{b^2} + {c^2} = 2m_a^2 + \frac{{{a^2}}}{2}\\
\,{a^2} + {c^2} = 2m_b^2 + \frac{{{b^2}}}{2}\\
\,{b^2} + {a^2} = 2m_c^2 + \frac{{{c^2}}}{2}
\end{array} \right\} \Rightarrow {a^2} + \,{b^2} + {c^2} = \frac{4}{3}({3^2} + {4^2} + {5^2})
= \frac{{200}}{3}$
$ \Rightarrow \,{a^2} + 2m_a^2 + \frac{{{a^2}}}{2} = \frac{{200}}{3} \Rightarrow a =
\frac{{10}}{3}cm$
Tương tự: $b = \frac{{4\sqrt {13} }}{3}\,\, ;\,\,c = \frac{{2\sqrt {73} }}{3}$
Theo định lý hàm số cosin:
$\cos A = \frac{{{b^2} + {c^2} - {a^2}}}{{2bc}} = \sqrt {\frac{{625}}{{949}}} \Rightarrow \cos
A > \frac{1}{{\sqrt 2 }} \Rightarrow A < {45^0}$
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