Ko biết thì đừng bình luận vô đây.

cho dãy tỉ số bằng nhau: 3a+b+2c/2a+c=a+3b+c/2b=a+2b+2c/b+c. tính giá trị biểu thức (a+b)(b+c)(c+a)/abc, với các mẫu số khác 0. Cái này cũng khó, nếu sai thì mong bạn thông cảm! 

Áp dụng t/c của DTSBN , ta có :

+, \(\dfrac{x}{a+2b+c}=\dfrac{y}{2a+b-c}=\dfrac{z}{4a-4b+c}\\ =\dfrac{x+2y+z}{a+2b+c+2\left(2a+b-c\right)+4a-4b+c}\\ =\dfrac{x+2y+z}{a+2b+c+4a+2b-2a-2c+4a-4b+c}\\ =\dfrac{x+2y+z}{\left(a+4a+4a\right)+\left(2b+2b-4b\right)+\left(c-2c+c\right)}\\ =\dfrac{x+2y+z}{9a}\left(1\right)\)

+, \(\dfrac{x}{a+2b+c}=\dfrac{y}{2a+b-c}=\dfrac{z}{4a-4b+c}\\ =\dfrac{2x+y-z}{2\left(a+2b+c\right)+2a+b-c-4a+4b+c}\\ =\dfrac{2x+y-z}{2a+4b+2c+2a+b-c-4a+4b+c}\\ =\dfrac{2x+y-z}{\left(2a+2a-4a\right)+\left(4b+b+4b\right)+\left(2c-c+c\right)}\\ =\dfrac{2x+y-z}{9b}\left(2\right)\)

+, \(\dfrac{x}{a+2b+c}=\dfrac{y}{2a+b-c}=\dfrac{z}{4a-4b+c}\\ =\dfrac{4x-4y+z}{4\left(a+2b+c\right)-4\left(2a+b-c\right)++4a-4b+c}\\ =\dfrac{4x-4y+z}{4a+8b+4c-8a-4b+4c+4a-4b+c}\\ =\dfrac{4x-4y+z}{\left(4a-8a+4a\right)+\left(8b-4b-4b\right)+\left(4c+4c+c\right)}\\ =\dfrac{4x-4y+z}{9c}\left(3\right)\)

Từ (1);(2) và (3) 

\(\Rightarrow\dfrac{x+2y+z}{9a}=\dfrac{2a+y-z}{9b}=\dfrac{4x-4y+z}{9c}\\ \Rightarrow\dfrac{x+2y+z}{9a}\cdot9=\dfrac{2a+y-z}{9b}\cdot9=\dfrac{4x-4y+z}{9c}\cdot9\\ \Rightarrow\dfrac{x+2y+z}{a}=\dfrac{2a+y-z}{b}=\dfrac{4x-4y+z}{c}\\ \Rightarrow\dfrac{a}{a+2y+z}=\dfrac{b}{2a+y-z}=\dfrac{c}{4x-4y+z}\left(đpcm\right)\)

Đặt \(\dfrac{x}{a+2b+c}=\dfrac{y}{2a+b-c}=\dfrac{z}{4a-4b+c}=k\)

\(\Rightarrow\left\{{}\begin{matrix}x=k\left(a+2b+c\right)\\y=k\left(2a+b-c\right)\\z=k\left(4a-4b+c\right)\end{matrix}\right.\)

\(\Rightarrow\dfrac{a}{x+2y+z}=\dfrac{a}{k\left(a+2b+c\right)+2k\left(2a+b-c\right)+k\left(4a-4b+c\right)}=\dfrac{a}{k.9a}=\dfrac{1}{9k}\)

Tượng tự:

\(\dfrac{b}{2x+y-z}=\dfrac{b}{9bk}=\dfrac{1}{9k}\) ; \(\dfrac{c}{4x-4y+z}=\dfrac{c}{9k.c}=\dfrac{1}{9k}\)

\(\Rightarrow\dfrac{a}{x+2y+z}=\dfrac{b}{2x+y-z}=\dfrac{c}{4x-4y+z}\)

Xét \(x+y+z=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}y+z=-x\\z+x=-y\\x+y=-z\end{matrix}\right.\)

\(\Rightarrow A=\left(2-1\right)\left(2-1\right)\left(2-1\right)=1\)

Xét \(x+y+z\ne0\) thì ta có:

\(\Rightarrow\left\{{}\begin{matrix}5x=y+z+3x\\5y=z+x+3y\\5z=x+y+3z\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2x=y+z\\2y=z+x\\2z=x+y\end{matrix}\right.\)

\(\Rightarrow A=\left(2+2\right)\left(2+2\right)\left(2+2\right)=64\)

Vậy \(\left[{}\begin{matrix}A=1\\A=64\end{matrix}\right.\)

Nếu bị lỗi thì bạn có thể xem đây nhé:

\(\dfrac{x}{2018}=\dfrac{y}{2019}=\dfrac{x-y}{-1};\dfrac{y}{2019}=\dfrac{z}{2020}=\dfrac{y-z}{-1};\dfrac{x}{2018}=\dfrac{z}{2020}=\dfrac{x-z}{-2}\\ \Leftrightarrow\dfrac{x-y}{-1}=\dfrac{y-z}{-1}=\dfrac{x-z}{-2}\\ \Leftrightarrow2\left(x-y\right)=2\left(y-z\right)=x-z\\ \Leftrightarrow\left(x-z\right)^3=8\left(x-y\right)^3=8\left(x-y\right)^2\left(x-y\right)=8\left(x-y\right)^2\left(y-z\right)\)

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

(y + z - x)/x = (z + x - y)/y = (x + y - z)/z = 1

--> y + z - x = x; z + x - y = y; x + y - z = z

--> y + z = 2x; z + x = 2y; x + y = 2z

Ta có: 

B = (x + y)/y.(y + z)/z.(z + x)/x

= 2z/y.2x/z.2y/x = 8