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Lời giải:
\(A=\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\)
\(A+3=\left(\frac{x}{y+z}+1\right)+\left(\frac{y}{z+x}+1\right)+\left(\frac{z}{x+y}+1\right)\)
\(A+3=\frac{x+y+z}{y+z}+\frac{x+y+z}{z+x}+\frac{x+y+z}{x+y}\)
\(A+3=2017\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)\)
\(A+3=2017.\frac{1}{672}=\frac{2017}{672}\)
\(\Rightarrow A=\frac{2017}{672}-3=\frac{1}{672}\)
Lời giải:
Áp dụng TCDTSBN:
$\frac{1}{x+y+z}=\frac{y+z+1}{x}=\frac{x+z+2}{y}=\frac{x+y-3}{z}=\frac{y+z+1+x+z+2+x+y-3}{x+y+z}=\frac{2(x+y+z)}{x+y+z}=2$
\(\Rightarrow \left\{\begin{matrix} x+y+z=\frac{1}{2}\\ y+z+1=2x\\ x+z+2=2y\\ x+y-3=2z\end{matrix}\right.\Rightarrow \left\{\begin{matrix} x+y+z=\frac{1}{2}\\ x+y+z+1=3x\\ x+y+z+2=3y\\ x+y+z-3=3z\end{matrix}\right.\)
\(\left\{\begin{matrix} \frac{1}{2}+1=3x\\ \frac{1}{2}+2=3y\\ \frac{1}{2}-3=3z\end{matrix}\right.\Rightarrow \left\{\begin{matrix} x=\frac{1}{2}\\ y=\frac{5}{6}\\ z=\frac{-5}{6}\end{matrix}\right.\)
Áp dụng t/c dtsbn ta có:
\(\dfrac{y+z+1}{x}=\dfrac{x+z+2}{y}=\dfrac{x+y-3}{z}=\dfrac{1}{x+y+z}=\dfrac{y+z+1+x+z+2+x+y-3}{x+y+z}=\dfrac{2\left(x+y+z\right)}{x+y+z}=2\)
\(\dfrac{1}{x+y+z}=2\Rightarrow2x+2y+2z=1\Rightarrow x+y+z=0,5\Rightarrow\left\{{}\begin{matrix}x+y=0,5-z\\y+z=0,5-x\\x+z=0,5-y\end{matrix}\right.\\ \dfrac{y+z+1}{x}=2\Rightarrow y+z+1=2x\Rightarrow0,5-x+1=2x\Rightarrow x=0,5\\ \dfrac{x+z+2}{y}=2\Rightarrow x+z+2=2y\Rightarrow0,5-y+2=2y\Rightarrow y=\dfrac{5}{6}\\ \dfrac{x+y-3}{z}=2\Rightarrow x+y-3=2z\Rightarrow0,5-z-3=2z\Rightarrow z=-\dfrac{5}{6}\)
\(\dfrac{x}{y}=\dfrac{y}{z}=\dfrac{z}{x}\\ \Rightarrow\left(\dfrac{x}{y}\right)^3=\dfrac{x}{y}.\dfrac{y}{z}.\dfrac{z}{x}=1\\ \Rightarrow\dfrac{x}{y}=1\\ \Rightarrow x=y\\ \Rightarrow y^{2017}-y^{2018}=0\\ \Rightarrow y^{2017}\left(1-y\right)=0\\ \Rightarrow\left[{}\begin{matrix}y=0\\y=1\end{matrix}\right.\)
Vì \(\dfrac{x}{y}=\dfrac{y}{z}=\dfrac{z}{x}\Rightarrow\left(\dfrac{x}{y}\right)^3=1\Leftrightarrow\dfrac{x}{y}=1\Rightarrow x=y\)
Mà \(x^{2017}-y^{2018}=1\Rightarrow y^{2017}\left(1-y\right)=1\)
\(\Rightarrow\left\{{}\begin{matrix}y^{2017}=1\\1-y=1\end{matrix}\right.\Rightarrow y=\left\{{}\begin{matrix}1\\0\end{matrix}\right.\)
Mà x = y
\(\Rightarrow x=\left\{{}\begin{matrix}1\\0\end{matrix}\right.\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\dfrac{x}{y+z+1}=\dfrac{y}{x+z+1}=\dfrac{z}{x+y-2}=x+y+z=\dfrac{x+y+z}{y+z+1+x+z+1+x+y-2}=\dfrac{x+y+z}{2x+2y+2z}=\dfrac{x+y+z}{2\left(x+y+z\right)}=\dfrac{1}{2}\)
\(\dfrac{x}{y+z+1}=\dfrac{1}{2}\Rightarrow y+z+1=2x\Rightarrow y+z=2x-1\left(1\right)\)
\(\dfrac{y}{x+z+1}=\dfrac{1}{2}\Rightarrow x+z+1=2y\Rightarrow x+z=2y-1\left(2\right)\)
\(\dfrac{z}{x+y-2}=\dfrac{1}{2}\Rightarrow x+y-2=2z\)
\(x+y+z=\dfrac{1}{2}\left(3\right)\)
Thay (1) vào (3) ta có:
\(x+y+z=\dfrac{1}{2}\\ \Rightarrow x+2x-1=\dfrac{1}{2}\\ \Rightarrow3x=\dfrac{3}{2}\\ \Rightarrow x=\dfrac{1}{2}\)
Thay (2) vào (3) ta có:
\(x+y+z=\dfrac{1}{2}\\ \Rightarrow y+2y-1=\dfrac{1}{2}\\ \Rightarrow3y=\dfrac{3}{2}\\ \Rightarrow y=\dfrac{1}{2}\)
Ta có:
\(x+y+z=\dfrac{1}{2}\\ \Rightarrow\dfrac{1}{2}+\dfrac{1}{2}+z=\dfrac{1}{2}\\ \Rightarrow z=-\dfrac{1}{2}\)
TH1: \(x+y+z=0\Rightarrow x=y=z=0\)
TH2: \(x+y+z\ne0\)
\(x+y+z=\dfrac{x}{y+z+1}=\dfrac{y}{x+z+1}=\dfrac{z}{x+y-2}=\dfrac{x+y+z}{2\left(x+y+z\right)}=\dfrac{1}{2}\)
\(\Leftrightarrow\)\(\left\{{}\begin{matrix}2x+2y+2z=1\\2x=y+z+1\\2y=x+z+1\\2z=x+y-2\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}2x+2y+2z=1\\2x+2y+2z=3y+3z+1\\2x+2y+2z=3x+3z+1\\2x+2y+2z=3x+3y-2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2x+2y+2z=1\\y+z=0\\x+z=0\\x+y=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2.1+2z=1\\y=-z\\x=-z\\x+y=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}z=-\dfrac{1}{2}\\x=\dfrac{1}{2}\\y=\dfrac{1}{2}\\\end{matrix}\right.\)
Vậy \(\left(x;y;z\right)=\left(0;0;0\right);\left(\dfrac{1}{2};\dfrac{1}{2};-\dfrac{1}{2}\right)\)
\(\Rightarrow\dfrac{z+y+1}{x}=\dfrac{x+z+1}{y}=\dfrac{x+y-2}{z}=\dfrac{2\left(x+y+z\right)}{x+y+z}=2=x+y+z\\ \Rightarrow\left\{{}\begin{matrix}z+y+1=2x\\x+z+1=2y\\x+y-2=2z\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}y+z=2x-1\\x+z=2y-1\\x+y=2z+2\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}2x-1=2-x\\2y-1=2-y\\2z+2=2-z\end{matrix}\right.\Rightarrow\left(x,y,z\right)=\left(1;1;0\right)\)
Áp dụng TCDTSBN ta có:
\(\dfrac{x+y+2017}{z}=\dfrac{y+z-2018}{x}=\dfrac{z+x+1}{y}=\dfrac{x+y+2017+y+z-2018+z+x+1}{z+x+y}=\dfrac{2x+2y+2z}{x+y+z}=\dfrac{2\left(x+y+z\right)}{x+y+z}=2\)
\(\dfrac{z+x+1}{y}=\dfrac{2}{x+y+z};\dfrac{z+x+1}{y}=2\\ \Rightarrow\dfrac{2}{x+y+z}=2\\ \Rightarrow x+y+z=1\)
\(\left\{{}\begin{matrix}\dfrac{x+y+2017}{z}=2\\\dfrac{y+z-2018}{x}=2\\\dfrac{z+x+1}{y}=2\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x+y+2017=2z\\y+z-2018=2x\\z+x+1=2y\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}x+y+z=3z-2017\\y+z+x=3x+2018\\z+x+y=3y-1\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}3z-2017=1\\3x+2018=1\\3y-1=1\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}3z=2018\\3x=-2017\\3y=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}z=\dfrac{2018}{3}\\x=\dfrac{-2017}{3}\\y=\dfrac{2}{3}\end{matrix}\right.\)
Vậy \(\left\{{}\begin{matrix}x=\dfrac{-2017}{3}\\y=\dfrac{2}{3}\\z=\dfrac{2018}{3}\end{matrix}\right.\)
Hình như là sai đề bn ak!