cho x^2+y^2+z^2=1 và x^5+y^5+z^5=1.
tính x^2020+y^2020+z^2020
K
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KN
8 tháng 10 2020
x2 + 2y2 + z2 - 2xy - 2y - 4z + 5 = 0
<=> ( x2 - 2xy + y2 ) + ( y2 - 2y + 1 ) + ( z2 - 4z + 4 ) = 0
<=> ( x - y )2 + ( y - 1 )2 + ( z - 2 )2 = 0
Vì \(\hept{\begin{cases}\left(x-y\right)^2\ge0\\\left(y-1\right)^2\ge0\\\left(z-2\right)^2\ge0\end{cases}}\forall x;y;z\)=> ( x - y )2 + ( y - 1 )2 + ( z - 2 )2\(\ge\)0\(\forall\)x ; y ; z
Dấu "=" xảy ra <=>\(\hept{\begin{cases}\left(x-y\right)^2=0\\\left(y-1\right)^2=0\\\left(z-2\right)^2=0\end{cases}}\)<=>\(\hept{\begin{cases}x=y=1\\z=2\end{cases}}\)( 1 )
Thay ( 1 ) vào A , ta được :
\(A=\left(1-1\right)^{2020}+\left(1-2\right)^{2020}+\left(2-3\right)^{2020}=0+1+1=2\)
Vậy A = 2
8 tháng 10 2020
Ta có: \(x^2+2y^2+z^2-2xy-2y-4z+5=0\)
\(\Leftrightarrow\left(x^2-2xy+y^2\right)+\left(y^2-2y+1\right)+\left(z^2-4z+4\right)=0\)
\(\Leftrightarrow\left(x-y\right)^2+\left(y-1\right)^2+\left(z-2\right)^2=0\)
Mà \(VT\ge0\left(\forall x,y,z\right)\) nên dấu "=" xảy ra khi:
\(\hept{\begin{cases}\left(x-y\right)^2=0\\\left(y-1\right)^2=0\\\left(z-2\right)^2=0\end{cases}}\Rightarrow\hept{\begin{cases}x=y=1\\z=2\end{cases}}\)
LD
30 tháng 10 2021
Ta có:
\(\frac{xy}{x+y}=\frac{yz}{y+z}=\frac{zx}{z+x}\rightarrow\frac{x+y}{xy}=\frac{y+z}{yz}=\frac{z+x}{zx}\)
\(\Rightarrow\frac{1}{x}+\frac{1}{y}=\frac{1}{y}+\frac{1}{z}=\frac{1}{z}+\frac{1}{x}\Rightarrow\frac{1}{x}=\frac{1}{y}=\frac{1}{z}\Rightarrow x=y=z\)
Thay tất cả giá trị x,y,z vào M ta được:
\(M=\frac{2020x^3+2020y^3+2020z^3}{x^3+y^3+z^3}+\frac{2021x^5+2021y^5}{x^5+y^5}\)
\(\Rightarrow M=\frac{2020\left(x^3+y^3+z^3\right)}{x^3+y^3+z^3}+\frac{2021\left(x^5+y^5\right)}{x^5+y^5}\)
\(\Rightarrow M=2020+2021=4041\)
30 tháng 10 2021
ĐKXĐ: \(\left\{{}\begin{matrix}2020-y^2\ge0\\2020-z^2\ge0\\2020-x^2\ge0\end{matrix}\right.\)
Ta có:
\(x\sqrt{2020-y^2}+y\sqrt{2020-z^2}+z\sqrt{2020-x^2}=3030\)
\(\Leftrightarrow2x\sqrt{2020-y^2}+2y\sqrt{2020-z^2}+2z\sqrt{2020-x^2}=6060\)
\(\Leftrightarrow2020-y^2-2x\sqrt{2020-y^2}+x^2+2020-z^2-2y\sqrt{2020-z^2}+y^2+2020-x^2-2z\sqrt{2020-x^2}+z^2=0\)
\(\Leftrightarrow\left(\sqrt{2020-y^2}-x\right)^2+\left(\sqrt{2020-z^2}-y\right)^2+\left(\sqrt{2020-x^2}-z\right)^2=0\)
\(\Leftrightarrow\left(\sqrt{2020-y^2}-x\right)^2=\left(\sqrt{2020-z^2}-y\right)^2=\left(\sqrt{2020-x^2}-z\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{2020-y^2}=x\\\sqrt{2020-z^2}=y\\\sqrt{2020-x^2}=z\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2020-y^2=x^2\\2020-z^2=y^2\\2020-x^2=z^2\end{matrix}\right.\)(vì \(x,y,z>0\))
\(\Leftrightarrow\left\{{}\begin{matrix}2020=x^2+y^2\\2020=y^2+z^2\\2020=z^2+x^2\end{matrix}\right.\)
\(\Rightarrow2\left(x^2+y^2+z^2\right)=3.2020\)
\(\Rightarrow x^2+y^2+z^2=3.1010=3030\)
\(\Rightarrow A=x^2+y^2+z^2=3030\)
Vậy \(A=3030\)
AH
Akai Haruma
Giáo viên
13 tháng 11 2023
Lời giải:
Áp dụng BĐT AM-GM:
\(x\sqrt{2020-y^2}+y\sqrt{2020-z^2}+z\sqrt{2020-x^2}\leq \frac{x^2+(2020-y^2)}{2}+\frac{y^2+(2020-z^2)}{2}+\frac{z^2+(2020-x^2)}{2}=3030\)Dấu "=" xảy ra khi:
\(\left\{\begin{matrix} x^2=2020-y^2\\ y^2=2020-z^2\\ z^2=2020-x^2\end{matrix}\right.\Rightarrow x=y=z=\sqrt{1010}\)
Khi đó:
$A=3(\sqrt{1010})^2=3030$
UI
1
7 tháng 8 2020
\(x^3+y^3+z^3+6=3\left(x^2+y^2+z^2\right)\Rightarrow\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)+3xyz+6=3\left(x^2+y^2+z^2\right)\)Mà x+y+z=3
\(\Rightarrow3\left(x^2+y^2+z^2-xy-xz-yz\right)+3xyz+6=3\left(x^2+y^2+z^2\right)\)
\(\Rightarrow x^2+y^2+z^2-xy-yz-xz+xyz+2=x^2+y^2+z^2\)
\(\Rightarrow xyz-xy-yz-xz+2=0\Rightarrow\left(xyz-xy\right)-\left(yz-y\right)-\left(xz-x\right)+\left(2-x-y\right)=0\)
\(\Rightarrow xy\left(z-1\right)-y\left(z-1\right)-x\left(z-1\right)+\left(2-3+z\right)=0\Rightarrow xy\left(z-1\right)-y\left(z-1\right)-x\left(z-1\right)+\left(z-1\right)=0\)
\(\Rightarrow\left(z-1\right)\left(xy-x-y+1\right)=0\Rightarrow\left(z-1\right)\left[\left(xy-x\right)-\left(y-1\right)\right]=0\Rightarrow\left(z-1\right)\left[x\left(y-1\right)-\left(y-1\right)\right]=0\)
\(\Rightarrow\left(z-1\right)\left(x-1\right)\left(y-1\right)=0\)
Suy ra có ít nhất 1 trong 3 số x,y,z bằng 1,khi đó A=0
Vậy A=0
link bài giải đây ạ => http://bblink.com/ghyht