ủa đây là toám lớp 1 hả anh

cauchy phần mẫu @@

(x+y+z)(xy+yz+zx)=xyz

x²y+xyz+zx²+xy²+y²z+xyz+xyz+yz²+z²x=xyz

(x²y+xy²)+(xyz+zx²)+(y²z+xyz)+(yz²+z²x)+xyz=xyz

xy(x+y)+zx(y+x)+yz(y+x)+z²(y+x)+xyz=xyz

(x+y)(xy+xz+yz+z²)+xyz=xyz

(x+y)[(xy+xz)+(yz+z²)]+xyz=xyz

(x+y)[x(y+z)+z(y+z)]+xyz=xyz

(x+y)(x+z)(y+z)+xyz=xyz

(x+y)(x+z)(y+z)=xyz-xyz

(x+y)(x+z)(y+z)=0

=>\(\left[{}\begin{matrix}x+y=0\\x+z=0\\y+z=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=-y\\x=-z\\y=-z\end{matrix}\right.\)

Với x=-z

=>VT= x²⁰¹⁵+y²⁰¹⁵+z²⁰¹⁵=(-z)²⁰¹⁵+z²⁰¹⁵+y²⁰¹⁵=y²⁰¹⁵

VP=(x+y+z)²⁰¹⁵=(-z+y+z)²⁰¹⁵=y²⁰¹⁵

Vậy x²⁰¹⁵+y²⁰¹⁵+z²⁰¹⁵=(x+y+z)²⁰¹⁵ với (x+y+z)(xy+yz+zx)=xyz

Đặt \(A=\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\)

Ta có:

\(x^2+xy+yz+zx=x+xyz=x\left(x+yz\right)\)

\(\Rightarrow\frac{x\left(x+yz\right)}{x}=\frac{x^2+xy+yz+zx}{x}\)

\(\Leftrightarrow x+yz=\frac{x^2+xy+yz+zx}{x}=\frac{\left(x^2+xy\right)+\left(yz+zx\right)}{x}=\frac{\left(x+z\right)\left(x+y\right)}{x}\)

\(\Rightarrow\sqrt{x+yz}=\sqrt{\frac{\left(x+y\right)\left(x+z\right)}{x}}\)

Vì x, y, z >0 nên áp dụng bất đẳng thức Bunhiacopxki cho 2 số dương, ta được:

\(\left(x+y\right)\left(x+z\right)\ge\left(\sqrt{x^2}.+\sqrt{yz}\right)^2\)

\(\Rightarrow\sqrt{\left(x+y\right)\left(x+z\right)}\ge x+\sqrt{yz}\)

\(\Rightarrow\sqrt{\frac{\left(x+y\right)\left(x+z\right)}{x}}\ge\frac{x+\sqrt{yz}}{\sqrt{x}}\)

Do đó \(\sqrt{x+yz}\ge\frac{x+\sqrt{yz}}{\sqrt{x}}\left(1\right)\)

Chứng minh tương tự, ta được:

\(\sqrt{y+xz}\ge\frac{y+\sqrt{xz}}{\sqrt{y}}\left(2\right)\)

Chứng minh tương tự, ta được:

\(\sqrt{z+xy}\ge\frac{z+\sqrt{xy}}{\sqrt{z}}\left(3\right)\)

Từ (1), (2) và (3), ta được:

\(\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\)\(\ge\frac{x+\sqrt{yz}}{\sqrt{x}}+\frac{y+\sqrt{zx}}{\sqrt{y}}+\frac{z+\sqrt{xy}}{\sqrt{z}}\)

\(\Leftrightarrow A\ge\sqrt{x}+\sqrt{\frac{yz}{x}}+\sqrt{y}+\sqrt{\frac{xz}{y}}+\sqrt{z}+\sqrt{\frac{xy}{z}}\)

\(\Leftrightarrow A\ge\sqrt{x}+\sqrt{y}+\sqrt{z}+\frac{yz+zx+xy}{\sqrt{xyz}}\)

 \(\Leftrightarrow A\ge\sqrt{x}+\sqrt{y}+\sqrt{z}+\frac{xyz}{\sqrt{xyz}}\)(vì \(xy+yz+zx=xyz\))

\(\Leftrightarrow A\ge\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{xyz}\)(điều phải chứng minh).

Dấu bằng xảy ra.

\(\Leftrightarrow\hept{\begin{cases}x=y=z>0\\xy+yz+zx=xyz\end{cases}}\Leftrightarrow x=y=z=3\)

Vậy với x, y, z là các số thực dương thỏa mãn xy + yz + zx =xyz thì:

\(\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{xyz}\).

\(\)

\(=\dfrac{xy\left(z-1\right)-y\left(z-1\right)-x\left(z-1\right)+\left(z-1\right)}{xy\left(z+1\right)+y\left(z+1\right)-x\left(z+1\right)-\left(z+1\right)}\\ =\dfrac{\left(z-1\right)\left(xy-y-x+1\right)}{\left(z+1\right)\left(xy+y-x-1\right)}=\dfrac{\left(z-1\right)\left(x-1\right)\left(y-1\right)}{\left(z+1\right)\left(x+1\right)\left(y-1\right)}=\dfrac{\left(z-1\right)\left(x-1\right)}{\left(z+1\right)\left(x+1\right)}\\ =\dfrac{\left(5003-1\right)\left(5001-1\right)}{\left(5003+1\right)\left(5001+1\right)}=\dfrac{5002\cdot5000}{5004\cdot5002}=\dfrac{5000}{5004}=\dfrac{1250}{1251}\)

13: 

xy(x+y)+yz(y+z)+xz(x+z)+2xyz 

= xy(x + y) + yz(y + z) + xyz + xz(x + z) + xyz 

= xy(x + y) + yz(y + z + x) + xz(x + z + y) 

= xy(x + y) + z(x + y + z)(y + x) 

= (x + y)(xy + zx + zy + z²) 

= (x + y)[x(y + z) + z(y + z)] 

= (x + y)(y + z)(z + x)