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20 tháng 6 2017

Cộng vế với vế của 3 đẳng thức đã cho ta được:

\(x+y+z-2\sqrt{y+2012}-2\sqrt{z-2013}-2\sqrt{x-2}=0\)

\(\Leftrightarrow\left(x-2-2\sqrt{x-2}+1\right)+\left(y+2012-2\sqrt{y+2012}+1\right)+\left(z-2013+2\sqrt{z-2013}+1\right)=0\)

\(\Leftrightarrow\left(\sqrt{x-2}-1\right)^2+\left(\sqrt{y+2012}-1\right)^2+\left(\sqrt{z-2013}-1\right)^2=0\)

\(\Rightarrow\left\{{}\begin{matrix}\left(\sqrt{x-2}-1\right)^2=0\\\left(\sqrt{y+2012}-1\right)^2=0\\\left(\sqrt{z-2013}-1\right)^2=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x-2}-1=0\\\sqrt{y+2012}-1=0\\\sqrt{z-2013}-1=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x-2}=1\\\sqrt{y+2012}=1\\\sqrt{z-2013}=1\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=-2011\\z=2014\end{matrix}\right.\)

Thay vào C ta được:

C = (3 - 4)2016 + (-2011 + 2012)2017 + (2014 - 2013)2018

C = 1 + 1 + 1 = 3

20 tháng 6 2017

THÊM

Cho x, y, z thõa mãn đồng thời: \(3x-2y-2\sqrt{y+2012}+1=0;3y-2z-2\sqrt{z-2013}+1=0;3z-2x-2\sqrt{x-2-2=0.}\)Tính \(C=\left(x-4\right)^{2016}+\left(y+2012\right)^{2017}+\left(z-2013\right)^{2018}\)

Ta có: \(3x^2+3y^2+4xy+2x-2y+2=0\)

\(\Leftrightarrow x^2+2x+1+y^2-2y+1+2x^2+4xy+2y^2=0\)

\(\Leftrightarrow\left(x+1\right)^2+\left(y-1\right)^2+2\left(x^2+2xy+y^2\right)=0\)

\(\Leftrightarrow\left(x+1\right)^2+\left(y-1\right)^2+2\left(x+y\right)^2=0\)

Ta có: \(\left(x+1\right)^2\ge0\forall x\)

\(\left(y-1\right)^2\ge0\forall y\)

\(2\left(x+y\right)^2\ge0\forall x,y\)

Do đó: \(\left(x+1\right)^2+\left(y-1\right)^2+2\left(x+y\right)^2\ge0\forall x,y\)

Dấu '=' xảy ra khi 

\(\left\{{}\begin{matrix}x+1=0\\y-1=0\\x+y=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-1\\y=1\\-1+1=0\left(đúng\right)\end{matrix}\right.\)

Thay x=-1 và y=1 vào biểu thức \(M=\left(x+y\right)^{2016}+\left(x+2\right)^{2017}+\left(y-1\right)^{2018}\), ta được: 

\(M=\left(-1+1\right)^{2016}+\left(-1+2\right)^{2017}+\left(1-1\right)^{2018}\)

\(=0^{2016}+1^{2017}+0^{2018}=1\)

Vậy: M=1

28 tháng 9 2016

Nếu\(a^3+b^3+c^3=3abc\Rightarrow\orbr{\begin{cases}a+b+c=0\\a=b=c\end{cases}}\)

Thật vậy:\(a+b+c=0\Rightarrow a+b=-c\\ \Rightarrow a^3+b^3+3ab\left(a+b\right)=-c^3\Rightarrow a^3+b^3+c^3=3abc\)

Tương tự \(a=b=c\Rightarrow\orbr{\begin{cases}3abc=3a^3\\a^3+b^3+c^3=3a^3\end{cases}\Rightarrow a^3+b^3+c^3=3abc}\)

Áp dụng ta có:\(\orbr{\begin{cases}xy+yz+zx=0\\xy=yz=zx\Rightarrow x=y=z\end{cases}}\)

Khi x=y=z,ta có P=(1+1)(1+1)(1+1)=8

Khi xy+yz+zx=0,ta có:\(xy+yz=-zx\)

Tương tự:\(yz+zx=-xy\)

               \(xy+zx=-yz\)

Ta có \(P=2+\frac{x+y}{z}+\frac{y+z}{x}+\frac{z+x}{y}=2+\frac{xz+yz}{z^2}+\frac{xy+xz}{x^2}+\frac{zy+xy}{y^2}\)\(=2-\left(\frac{z}{x}+\frac{x}{y}+\frac{y}{z}\right)\)\(=2-\frac{xy+yz+zx}{xyz}=2-\frac{0}{xyz}=2\)

Vậy P=8 khi x=y=z

      P=2 khi xy+yz+zx=0

28 tháng 9 2016

kho nhi

8 tháng 8 2016

Bạn xem lại đề nhé :)

Thay 1 bằng xy + yz + zx được : 

\(1+y^2=xy+yz+zx+y^2=x\left(y+z\right)+y\left(y+z\right)=\left(x+y\right)\left(y+z\right)\)

Tương tự : \(1+x^2=\left(x+y\right)\left(x+z\right)\)\(1+z^2=\left(x+z\right)\left(z+y\right)\)

Suy ra \(Q=x\sqrt{\frac{\left(x+y\right)\left(y+z\right).\left(x+z\right)\left(z+y\right)}{\left(x+y\right)\left(x+z\right)}}+y\sqrt{\frac{\left(x+y\right)\left(x+z\right).\left(z+x\right)\left(z+y\right)}{\left(x+y\right)\left(y+z\right)}}+z\sqrt{\frac{\left(x+y\right)\left(x+z\right).\left(x+y\right)\left(y+z\right)}{\left(x+z\right)\left(z+y\right)}}\)

\(=x\sqrt{\left(y+z\right)^2}+y\sqrt{\left(x+z\right)^2}+z\sqrt{\left(x+y\right)^2}=x\left|y+z\right|+y\left|x+z\right|+z\left|x+y\right|\)

\(=2\left(xy+yz+zx\right)=2\)(vì x,y,z > 0)

Ta có : \(4x^2+2y^2+2z^2-4xy-4zx+2yz-6y-10z+34=0\)

\(\Rightarrow\left(4x^2+y^2+z^2-4xy-4zx+2yz\right)+\left(y^2-6y+9\right)+\left(z^2-10z+25\right)=0\)

\(\Rightarrow\left(2x-y-z\right)^2+\left(y-3\right)^2+\left(z-5\right)^2=0\)

Vì \(\hept{\begin{cases}\left(2x-y-z\right)^2\ge0\forall x,y,z\\\left(y-3\right)^2\ge0\forall y\\\left(z-5\right)^2\ge0\forall z\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}\left(2x-y-z\right)^2=0\\\left(y-3\right)^2=0\\\left(z-5\right)^2=0\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}2x-y-z=0\\y-3=0\\z-5=0\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}2x-3-5=0\\y=3\\z=5\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}2x=8\\y=3\\z=5\end{cases}}\)\(\Rightarrow\hept{\begin{cases}x=4\\y=3\\z=5\end{cases}}\left(1\right)\)

Lại có : \(S=\left(x-4\right)^{2017}+\left(y-4\right)^{2017}+\left(z-4\right)^{2017}\)

Thay \(\left(1\right)\)vào \(S\),ta được :

\(S=0^{2017}+\left(-1\right)^{2017}+1^{2017}\)

    \(=0-1+1=0\)

Vậy \(S=0\)

NV
6 tháng 1 2019

Ta có: \(x+2y+3x=0\Leftrightarrow x=-\left(2y+3z\right)\)

Lại có: \(2xy+6yz+3xz=0\Leftrightarrow x\left(2y+3z\right)+6yz=0\)

\(\Leftrightarrow-\left(2y+3z\right)\left(2y+3z\right)+6yz=0\Leftrightarrow-\left(2y+3z\right)^2+6yz=0\)

\(\Leftrightarrow\left(2y+3z\right)^2-6yz=0\Leftrightarrow4y^2+12yz+9z^2-6yz=0\)

\(\Leftrightarrow4y^2+6yz+9z^2=0\Leftrightarrow\left(2y+\dfrac{3z}{2}\right)^2+\dfrac{27z^2}{4}=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(2y+\dfrac{3z}{2}\right)^2=0\\\dfrac{27z^2}{4}=0\end{matrix}\right.\) \(\Rightarrow y=z=0\Rightarrow x=0\)

\(\Rightarrow S=\dfrac{\left(-1\right)^{2019}-1^{2017}+\left(-1\right)^{2015}}{1^{2018}+2.0^{2016}+0^{2014}+2}=\dfrac{-1-1+-1}{1+0+0+2}=\dfrac{-3}{3}=-1\)