Dat \(\left(\frac{a}{b};\frac{b}{c};\frac{c}{a}\right)=\left(x;y;z\right)\)
\(\Rightarrow xyz=1\)
\(\Sigma_{cyc}\frac{1}{\frac{a}{b}+\frac{c}{a}+1}=\Sigma_{cyc}\frac{1}{x+y+1}\)
We need to prove:
\(\Sigma_{cyc}\frac{1}{x+y+1}\le1\)
\(\Leftrightarrow\Sigma_{cyc}\frac{x+y}{x+y+1}\ge2\left(M\right)\)
We have:
\(VT_M\ge\frac{\left(\Sigma_{cyc}\sqrt{x+y}\right)^2}{2\Sigma_{cyc}x+3}\)
Now we need to prove
\(\frac{\left(\Sigma_{cyc}\sqrt{x+y}\right)^2}{2\Sigma_{cyc}x+3}\ge2\)
\(\Leftrightarrow\Sigma_{cyc}\sqrt{\left(x+y\right)\left(y+z\right)}\ge\Sigma_{cyc}x+3\left(M_1\right)\)
Consider:
\(VT_{M_1}=\sqrt{\left(x+y\right)\left(y+z\right)}\ge x+y+z+xy+yz+zx\)
Now we need to prove:
\(x+y+z+xy+yz+zx\ge x+y+z+3\)
\(xy+yz+zx\ge3\) (Not fail with xyz=1)
Dau '=' xay ra khi \(\hept{\begin{cases}a=b=c=1\\x=y=z=1\end{cases}}\)
Mấy cái kí hiệu kia là gì v bạn
