`A=(3x+2)/(x-1)`

`=(3x-3+5)/(x-1)`

`=(3(x-1)+5)/(x-1)`

`=3 + 5/(x-1)`

Để `A` nguyên thì `5/(x-1)` nguyên

`=> x-1 in Ư(5)`

`=>x-1 in {-5;-1;1;5}`

`=>x in {-4;0;2;6}`

\(\dfrac{x+\sqrt{xy}}{y+\sqrt{xy}}\\ =\dfrac{\sqrt{x^2}+\sqrt{xy}}{\sqrt{y^2}+\sqrt{xy}}\\ =\dfrac{\sqrt{x}\left(\sqrt{x}+y\right)}{\sqrt{y}\left(\sqrt{x}+y\right)}\\ =\dfrac{\sqrt{x}}{\sqrt{y}}\)

\(P=\left(\dfrac{x-\sqrt{x}+2}{x-\sqrt{x}-2}-\dfrac{x}{x-2\sqrt{x}}\right):\dfrac{1-\sqrt{x}}{x-2\sqrt{x}}\\ =\left(\dfrac{x-\sqrt{x}+2}{x-2\sqrt{x}+\sqrt{x}-2}-\dfrac{x}{\sqrt{x}\left(\sqrt{x}-2\right)}\right):\dfrac{1-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-2\right)}\\ =\left(\dfrac{x-\sqrt{x}+2}{\sqrt{x}\left(\sqrt{x}-2\right)+\left(\sqrt{x}-2\right)}-\dfrac{x}{\sqrt{x}\left(\sqrt{x}-2\right)}\right).\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)}{1-\sqrt{x}}\)

\(=\left(\dfrac{x-\sqrt{x}+2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}-\dfrac{x}{\sqrt{x}\left(\sqrt{x}-2\right)}\right).\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)}{1-\sqrt{x}}\\ =\left(\dfrac{\sqrt{x}\left(x-\sqrt{x}+2\right)}{\sqrt{x}\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}-\dfrac{x\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}\right).\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)}{1-\sqrt{x}}\\ =\left(\dfrac{x\sqrt{x}-x+2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}-\dfrac{x\sqrt{x}+x}{\sqrt{x}\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}\right).\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)}{1-\sqrt{x}}\)

\(=\dfrac{x\sqrt{x}-x+2\sqrt{x}-x\sqrt{x}-x}{\sqrt{x}\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}.\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)}{1-\sqrt{x}}\\ =\dfrac{-2x+2\sqrt{x}}{\sqrt{x}+1}.\dfrac{1}{1-\sqrt{x}}\\ =\dfrac{-2\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}+1}.\dfrac{1}{-\left(\sqrt{x}-1\right)}\\ =\dfrac{2\sqrt{x}}{\sqrt{x}+1}\)

\(\dfrac{4}{15}.\dfrac{3}{17}+\dfrac{12}{35}+\dfrac{4}{15}.\dfrac{-20}{17}+\dfrac{23}{35}\\ =\dfrac{4}{15}\cdot\left(\dfrac{3}{17}+\dfrac{-20}{17}\right)+\left(\dfrac{12}{35}+\dfrac{23}{35}\right)\\ =\dfrac{4}{15}.\dfrac{-17}{17}+\dfrac{35}{35}\\ =\dfrac{4}{15}.\left(-1\right)+1\\ =\dfrac{-4}{15}+1\\ =\dfrac{-4}{15}+\dfrac{15}{15}\\ =\dfrac{11}{15}\)

Cho `ΔABC` ngoại tiếp `(I)`. `E,F` là tiếp điểm của `(I)` với `AB,AC` . Đường thẳng `BI,CI` cắt `EF` tại `M,N` . Chứng minh: `F,M,C,I` cùng thuộc 1 đường tròn

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

`x/3 = y/4 = z/6 = (x+y+z)/(3+4+6) = 52/13 =4`

`x/3 =4=>x=3.4=12

`y/4 = 4=>y=4.4=16`

`z/6 =4=>z=4.6=24`

Cho `ΔABC` ngoại tiếp `(I)`. `E,F` là tiếp điểm của `(I)` với `AB,AC` . Đường thẳng `BI,CI` cắt `EF` tại `M,N` . Chứng minh: `B,C,N,M` cùng thuộc 1 đường tròn