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Từ giả thiết suy ra (ay+bx)/xy = (bz+cy)/yz =(cx+az)/xz hay a/x =b/y =c/z.
dặt x/a =y=b =z/c =k suy ra x =ak; y=bk; z=ck. thay vào biểu thức bài cho tìm được k=1/2
vậy x =a/2; y=b/2; z=c/2
\(\frac{xy}{ay+bx}\)=\(\frac{yz}{bz+cy}\)=\(\frac{zx}{cx+az}\left(1\right)\)
\(\Rightarrow\)\(\frac{xyz}{ayz+bxz}\)=\(\frac{xyz}{bzx+cyx}\)=\(\frac{zyx}{cxy+azy}\)
\(\Rightarrow\)\(ayz+bxz=bzx+cyx=cxy+azy\)
\(\Rightarrow\)\(\hept{\begin{cases}ayz+bxz=bxz+cyx\\bzx+cyx=cxy+azy\\ayz+bxz=cxy+azy\end{cases}}\Rightarrow\hept{\begin{cases}ayz=cyx\\bzx=azy\\bxz=cxy\end{cases}}\)\(\Rightarrow\hept{\begin{cases}az=cx\\bx=ay\\bz=cy\end{cases}\left(2\right)}\)
thay (2) vào (1)
\(\Rightarrow\)\(\frac{xy}{2ay}\)=\(\frac{yz}{2bz}\)=\(\frac{zx}{2cx}\)
\(\Rightarrow\)\(\frac{x}{2a}=\frac{y}{2b}=\frac{z}{2c}\)\(=\frac{x^2+y^2+z^2}{a^2+b^2+c^2}\left(3\right)\)
\(\Rightarrow\left(\frac{x}{2a}\right)^2=\left(\frac{y}{2b}\right)^2=\left(\frac{z}{2c}\right)^2\)
\(\Rightarrow\text{}\text{}\)\(\frac{x^2}{4a^2}=\frac{y^2}{4b^2}=\frac{z^2}{4c^2}\)
theo quy luật của dãy số bằng nhau, nên
\(\frac{x^2}{4a^2}=\frac{y^2}{4b^2}=\frac{z^2}{4c^2}=\)\(\frac{x^2+y^2+z^2}{4a^2+4b^2+4c^2}=\frac{\left(x^2+y^2+z^2\right)}{4\left(a^2+b^2+c^2\right)}=\frac{1}{4}\left(4\right)\)
từ (3) và (4)
\(\Rightarrow\)\(\frac{x}{2a}=\frac{y}{2b}=\frac{z}{2c}=\frac{1}{4}\)
\(\Rightarrow\hept{\begin{cases}x=\frac{a}{2}\\y=\frac{b}{2}\\c=\frac{c}{2}\end{cases}}\)
\(A=\frac{x}{xy+x+1}+\frac{y}{yz+y+1}+ \frac{z}{xz+z+1}\)
\(=\frac{x}{xyz+xy+x}+\frac{y}{yz+y+1}+\frac{z}{xz+z+1}\)
\(=\frac{1}{yz+y+1}+\frac{y}{yz+y+1}+\frac{z}{xz+z+1}\)
\(=\frac{y+1}{yz+y+1}+\frac{z}{xz+z+1}\)
\(=\frac{xyz+y}{xyz+yz+y}+\frac{z}{xz+z+1 }\)
\(=\frac{xz+1}{xz+z+1}+\frac{z}{xz+z+1}\)
\(=\frac{xz+z+1}{xz+z+1}=1\)
#Carrot
Sửa lại đề : \(A=\frac{yz}{x^2+2yz}+\frac{xz}{y^2+2xz}+\frac{xy}{z^2+2xy}\)
Ta có : \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)
\(\Rightarrow xy+yz+xz=0\)
\(\Rightarrow\hept{\begin{cases}xy=-yz-xz\\yz=-xy-xz\\zx=-yz-xy\end{cases}\left(1\right)}\)
Thay (1) vào A, ta có :
\(A=\frac{yz}{x^2+2yz}+\frac{xz}{y^2+2xz}+\frac{xy}{z^2+2xy}\)
\(=\frac{yz}{x^2+yz-xy-xz}+\frac{xz}{y^2+xz-yz-xy}+\frac{xy}{z^2+xy-yz-xz}\)
\(=\frac{yz}{\left(x-y\right)\left(x-z\right)}+\frac{xz}{\left(y-z\right)\left(y-x\right)}+\frac{xy}{\left(z-y\right)\left(z-x\right)}\)
\(=\frac{yz}{\left(x-y\right)\left(x-z\right)}-\frac{xz}{\left(y-z\right)\left(x-y\right)}+\frac{xy}{\left(z-y\right)\left(z-x\right)}\)
\(=\frac{yz\left(y-z\right)-xz\left(x-z\right)+xy\left(x-y\right)}{\left(x-y\right)\left(y-z\right)\left(x-z\right)}\)
\(=\frac{\left(x-y\right)\left(y-z\right)\left(x-z\right)}{\left(x-y\right)\left(y-z\right)\left(x-z\right)}=1\)
\(\text{Σ}\frac{x^2}{\sqrt[3]{x^3+8}}=\text{Σ}\frac{x^2}{\sqrt[3]{\left(x+2\right)\left(x^2-2x+4\right)}}\ge\text{Σ}\frac{x^2}{\frac{x+2+x^2-2x+4}{2}}=\text{2}\left(Σ\frac{x^2}{x^2-x+6}\right)\)
Áp dụng BDT Cauchy-Schwarz:
\(VT\ge2\frac{\left(x+y+z\right)^2}{x^2+y^2+z^2-x-y-z+18}\)
Áp dụng BDT: \(9=3\left(xy+yz+xz\right)\le\left(x+y+z\right)^2\Rightarrow x+y+z\ge3\)
\(\Rightarrow VT\ge2\frac{\left(x+y+z\right)^2}{x^2+y^2+z^2-3+18}=2\frac{\left(x+y+z\right)^2}{x^2+y^2+z^2+15}=2\frac{\left(x+y+z\right)^2}{\left(x+y+z\right)^2+3\left(xy+yz+xz\right)}\)
\(\ge2\frac{\left(x+y+z\right)^2}{2\left(x+y+z\right)^2}=1\)
Dấu = xảy ra khi x=y=z=1
Ta có \(\frac{xy}{x+y}=\frac{yz}{y+z}=\frac{xz}{x+z}\)
=> \(\frac{xyz}{xz+yz}=\frac{xyz}{xy+xz}=\frac{xyz}{xy+yz}\)
=> \(xz+yz=xy+xz=xy+yz\)(vì x ; y ;z \(\ne0\Leftrightarrow xyz\ne0\))
=> \(\hept{\begin{cases}xz+yz=xy+xz\\xy+xz=xy+yz\\xz+yz=xy+yz\end{cases}}\Rightarrow\hept{\begin{cases}yz=xy\\xz=yz\\xz=xy\end{cases}}\Rightarrow\hept{\begin{cases}z=x\\x=y\\y=z\end{cases}}\Rightarrow x=y=z\)
Khi đó M = \(\frac{x^2+y^2+z^2}{xy+yz+zx}=\frac{x^2+y^2+z^2}{x^2+y^2+z^2}=1\left(\text{vì }x=y=z\right)\)
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