a) \(\overrightarrow{IA}+2\overrightarrow{IB}=\overrightarrow{BA}+3\overrightarrow{IB}=\overrightarrow{0}\Rightarrow\overrightarrow{BI}=\frac{1}{3}\overrightarrow{BA}\)

\(\overrightarrow{CI}=\overrightarrow{CB}+\overrightarrow{BI}=\overrightarrow{CB}+\frac{1}{3}\overrightarrow{BA}=\overrightarrow{CB}+\frac{1}{3}\left(\overrightarrow{CA}-\overrightarrow{CB}\right)=\frac{2}{3}\overrightarrow{CB}+\frac{1}{3}\overrightarrow{CA}\)

\(\overrightarrow{JB}=x\overrightarrow{JC}\Rightarrow\overrightarrow{CB}-\overrightarrow{CJ}=x\overrightarrow{JC}\Rightarrow\overrightarrow{CB}=\left(x-1\right)\overrightarrow{JC}\Rightarrow\overrightarrow{CJ}=\frac{1}{1-x}\overrightarrow{CB}\)

b) \(\overrightarrow{IJ}=\overrightarrow{CJ}-\overrightarrow{CI}=\frac{1}{1-x}\overrightarrow{CB}-\left(\frac{2}{3}\overrightarrow{CB}+\frac{1}{3}\overrightarrow{CA}\right)=\frac{2x+1}{3\left(1-x\right)}\overrightarrow{CB}-\frac{1}{3}\overrightarrow{CA}\)

c) Dễ có \(\overrightarrow{CG}=\frac{2}{3}\left(\overrightarrow{CB}+\overrightarrow{CA}\right)\). Để \(\overrightarrow{IJ}\)//\(\overrightarrow{CG}\) thì :

\(\frac{\frac{2}{3}}{\frac{2x+1}{3\left(1-x\right)}}=\frac{\frac{2}{3}}{-\frac{1}{3}}\Leftrightarrow\frac{1-x}{2x+1}=-1\Rightarrow2x+1=x-1\Leftrightarrow x=-2\)

Vậy \(x=-2\)tức \(\overrightarrow{JB}=-2\overrightarrow{JC}\)thì IJ // CG.

* Nhận xét: Nếu \(\overrightarrow{u}=x\overrightarrow{a}+y\overrightarrow{b};\overrightarrow{v}=m\overrightarrow{a}+n\overrightarrow{b}\)thì \(\overrightarrow{u}\)//\(\overrightarrow{v}\)\(\Leftrightarrow\frac{x}{m}=\frac{y}{n}.\)

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\(\frac{5}{x}+\frac{4}{x+1}=\frac{3}{x+2}+\frac{2}{x+3}\)

\(\Leftrightarrow\frac{5\left(x+1\right)+4x}{x\left(x+1\right)}=\frac{3\left(x+3\right)+2\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}\)

\(\Leftrightarrow\frac{5x+5+4x}{x^2+x}=\frac{3x+9+2x+4}{x^2+5x+6}\)

\(\Leftrightarrow\frac{9x+5}{x^2+x}=\frac{5x+13}{x^2+5x+6}\)

\(\Leftrightarrow\left(9x+5\right)\left(x^2+5x+6\right)=\left(5x+13\right)\left(x^2+x\right)\)

\(\Leftrightarrow9x^3+45x^2+54x+5x^2+25x+30=5x^3+5x^2+13x^2+13x\)

\(\Leftrightarrow9x^3+50x^2+79x+30=5x^3+18x^2+13x\)

\(\Leftrightarrow9x^3-5x^3+50x^2-18x^2+79x-13x+30=0\)

\(\Leftrightarrow4x^3+32x^2+66x+30=0\)

\(\Leftrightarrow2x^3+16x^2+33x+15=0\)

\(\Leftrightarrow\left(x-5\right)\left(x+2,3660\right)\left(x+0,6340\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=5\\x\approx2,3660\end{cases}or_{ }x\approx0,6340}\)

Nghiệm là -5 thôi nha, phần còn lại khác 0 nên loại

\(A=\left(\frac{2}{\sqrt{x}-2}+\frac{3}{2\sqrt{x}+1}-\frac{5\sqrt{x}-7}{2x-3\sqrt{x}-2}\right):\)\(\frac{2\sqrt{x}+3}{5x-10\sqrt{x}}\)

\(=\left(\frac{2}{\sqrt{x}-2}+\frac{3}{2\sqrt{x}+1}-\frac{5\sqrt{x}-7}{\left(\sqrt{x}-2\right)\left(2\sqrt{x}+1\right)}\right)\)\(:\frac{2\sqrt{x}+3}{5x-10\sqrt{x}}\)

\(=\frac{2\left(2\sqrt{x}+1\right)+3\left(\sqrt{x}-2\right)-5\sqrt{x}+7}{\left(\sqrt{x}-2\right)\left(2\sqrt{x}+1\right)}\)\(:\frac{2\sqrt{x}+3}{5\sqrt{x}\left(\sqrt{x}-2\right)}\)

\(=\frac{4\sqrt{x}+2+3\sqrt{x}-6-5\sqrt{x}+7}{\left(\sqrt{x}-2\right)\left(2\sqrt{x}+1\right)}\)\(.\frac{5\sqrt{x}\left(\sqrt{x}-2\right)}{2\sqrt{x}+3}\)

\(=\frac{2\sqrt{x}+3}{2\sqrt{x}+1}.\frac{5\sqrt{x}}{2\sqrt{x}+3}=\frac{5\sqrt{x}}{2\sqrt{x}+1}\)

\(A\in Z\Leftrightarrow\frac{5\sqrt{x}}{2\sqrt{x}+1}\in Z\Leftrightarrow\frac{10\sqrt{x}}{2\sqrt{x}+1}\in Z\)

\(\Rightarrow\frac{10\sqrt{x}+5-5}{2\sqrt{x}+1}\in Z\Leftrightarrow5-\frac{5}{2\sqrt{x}+1}\in Z\)

\(\Rightarrow\frac{5}{2\sqrt{x}+1}\in Z\Rightarrow2\sqrt{x}+1\inƯ_5\)

Mà \(Ư_5=\left\{\pm1;\pm5\right\}\)

Nhưng \(2\sqrt{x}+1\ge1\)

\(\Rightarrow\orbr{\begin{cases}2\sqrt{x}+1=1\\2\sqrt{x}+1=5\end{cases}\Rightarrow\orbr{\begin{cases}2\sqrt{x}=0\\2\sqrt{x}=4\end{cases}}}\)

\(\Rightarrow\orbr{\begin{cases}\sqrt{x}=0\\\sqrt{x}=2\end{cases}\Rightarrow\orbr{\begin{cases}x=0\\x=4\end{cases}}}\)

Vậy \(x\in\left\{0;4\right\}\)