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\(ab+bc+ca=abc\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\)
Đặt \(\frac{1}{a}=x;\frac{1}{b}=y;\frac{1}{c}=z\)
\(\frac{a}{bc\left(a+1\right)}=\frac{\frac{1}{x}}{\frac{1}{y}\cdot\frac{1}{z}\left(\frac{1}{x}+1\right)}=\frac{xyz}{x\left(x+1\right)}=\frac{yz}{x+1}\)
Tươn tự rồi cộng vế theo vế:
\(A=\frac{xy}{z+1}+\frac{yz}{x+1}+\frac{zx}{y+1}\le\frac{\left(x+y\right)^2}{4\left(z+1\right)}+\frac{\left(y+z\right)^2}{4\left(x+1\right)}+\frac{\left(z+x\right)^2}{4\left(y+1\right)}\)
Đặt \(x+y=p;y+z=q;z+x=r\Rightarrow p+q+r=2\)
\(A\le\Sigma\frac{\left(x+y\right)^2}{4\left(z+1\right)}=\Sigma\frac{\left(x+y\right)^2}{4\left[\left(z+y\right)+\left(z+x\right)\right]}=\frac{p^2}{4\left(q+r\right)}+\frac{r^2}{4\left(p+q\right)}+\frac{q^2}{4\left(p+r\right)}\)
Sau khi đổi biến,cô si thì em ra thế này.Ai đó giúp em với :)
Ta có: \(ab+bc+ca=abc\)
\(\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\)
Đặt: \(A=\frac{a}{bc\left(a+1\right)}+\frac{b}{ca\left(b+1\right)}+\frac{c}{ab\left(c+1\right)}\)
\(\Rightarrow A=\frac{\frac{1}{b}.\frac{1}{c}}{1+\frac{1}{a}}+\frac{\frac{1}{c}.\frac{1}{a}}{1+\frac{1}{b}}+\frac{\frac{1}{b}.\frac{1}{a}}{1+\frac{1}{c}}\)
Đặt: \(\hept{\begin{cases}x=\frac{1}{a}\\y=\frac{1}{b}\\z=\frac{1}{c}\end{cases}}\Rightarrow x+y+z=1\)
\(A=\frac{xy}{z+1}+\frac{yz}{x+1}+\frac{zx}{y+1}\)
Ta có: \(\frac{xy}{z+1}=\frac{xy}{\left(z+x\right)+\left(z+y\right)}\le\frac{1}{4}\left(\frac{xy}{x+z}+\frac{xy}{y+z}\right)\)
Chứng minh tương tự ta được:
\(\frac{yz}{x+1}\le\frac{yz}{x+y}+\frac{yz}{x+z}\)
\(\frac{zx}{y+1}\le\frac{zx}{x+y}+\frac{zx}{y+z}\)
Cộng vế với vế:
\(\Rightarrow A\le\frac{1}{4}\left(x+y+z\right)=\frac{1}{4}\left(đpcm\right)\)
ÁP dụng BĐT AM-Gm ta có:
\(Σ\frac{a^2}{\left(ab+2\right)\left(2ab+1\right)}\ge\frac{4}{9}\cdotΣ\frac{a^2}{\left(ab+1\right)^2}\)
ĐẶt \(a=\frac{x}{y};b=\frac{y}{z};c=\frac{z}{x}\) thì cần cm
\(Σ\frac{a^2}{\left(ab+1\right)^2}=Σ\left(\frac{xz}{y\left(x+z\right)}\right)^2\ge\frac{3}{4}\)
\(Σ\left(\frac{xz}{y\left(x+z\right)}\right)^2\ge\frac{1}{3}\left(\frac{xz}{y\left(x+z\right)}\right)^2\)
Theo C-S \(Σ\frac{xz}{y\left(x+z\right)}=\frac{\left(xz\right)^2}{xyz\left(x+z\right)}\ge\frac{\left(Σxy\right)^2}{2xy\left(Σx\right)}\ge\frac{3}{2}\)
\(\frac{1}{3}\cdot\left(Σ\frac{xz}{y\left(x+z\right)}\right)^2\ge\frac{1}{3}\cdot\frac{9}{4}=\frac{3}{4}\)
Đúng hay ta có ĐPCM xyar ra khi a=b=c=1
\(P=\frac{a^3}{\left(a+1\right)\left(b+1\right)}+\frac{b^3}{\left(b+1\right)\left(c+1\right)}+\frac{c^3}{\left(c+1\right)\left(a+1\right)}-1\)
