\(a^2b^2c^2+\left(a+1\right)\left(1+b\right)\left(1+c\right)\ge a+b+c+ab+bc+ca+3\)

\(\Leftrightarrow\left(abc\right)^2+abc-2\ge0\Leftrightarrow\left(abc+2\right)\left(abc-1\right)\ge0\Leftrightarrow abc\ge1\)

Áp dụng BĐT Cosi ta có:

\(\frac{a^3}{\left(b+2c\right)\left(2c+3a\right)}+\frac{b+2c}{45}+\frac{2c+3a}{75}\ge3\sqrt[3]{\frac{a^3}{\left(b+2c\right)\left(2c+3b\right)}\cdot\frac{b+2c}{45}\cdot\frac{2c+3a}{75}}=\frac{a}{5}\left(1\right)\)

Tương tự ta có: \(\hept{\begin{cases}\frac{b^3}{\left(c+2a\right)\left(2a+3b\right)}+\frac{c+2a}{45}+\frac{2a+3b}{75}\ge\frac{b}{5}\left(2\right)\\\frac{c^3}{\left(a+2b\right)\left(2b+3c\right)}+\frac{a+2b}{45}+\frac{2b+3c}{75}\ge\frac{c}{5}\left(3\right)\end{cases}}\)

Từ (1)(2)(3) ta có:

\(P+\frac{2\left(a+b+c\right)}{15}\ge\frac{a+b+c}{5}\Leftrightarrow P\ge\frac{1}{15}\left(a+b+c\right)\)

Mà \(a+b+c\ge3\sqrt[3]{abc}\Rightarrow S\ge\frac{1}{5}\)

Dấu "=" xảy ra <=> a=b=c=1

CHÚC BAN HỌC GIỎI

đây\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có : 

\(\frac{2b+c-a}{a}=\frac{2c-b+a}{b}=\frac{2a+b-c}{c}=\frac{2a+2b+2c}{a+b+c}=\frac{2\left(a+b+c\right)}{a+b+c}=2\)

Do đó : 

\(\frac{2b+c-a}{a}=2\)\(\Rightarrow\)\(c=3a-2b\)\(;\)\(2b=3a-c\)\(\left(1\right)\)

\(\frac{2c-b+a}{b}=2\)\(\Rightarrow\)\(a=3b-2c\)\(;\)\(2c=3b-a\)\(\left(2\right)\)

\(\frac{2a+b-c}{c}=2\)\(\Rightarrow\)\(b=3c-2a\)\(;\)\(2a=3c-b\)\(\left(3\right)\)

Thay (1), (2) và (3) vào \(P=\frac{\left(3a-2b\right)\left(3b-2c\right)\left(3c-2a\right)}{\left(3a-c\right)\left(3b-a\right)\left(3c-b\right)}\) ta được : 

\(P=\frac{c.a.b}{2b.2c.2a}=\frac{abc}{8abc}=\frac{1}{8}\)

Vậy \(P=\frac{1}{8}\)

Chúc bạn học tốt ~ 

Phùng Minh Quân sai nha nếu a+b+c = 0 thì a+b+c / 2(a+b+c) thì nó không bằng 1/2 đc mà nó bằng 0

\(\frac{2b+c-a}{a}=\frac{2c-b+a}{b}=\frac{2a+b-c}{c}=\frac{2b+c-a+2c-b+a+2a+b-c}{a+b+c}=\)

\(=\frac{2a+2b+2c}{a+b+c}=2\)

+ Từ \(\frac{2b+c-a}{a}=2\Rightarrow2b+c-a=2a\Rightarrow3a-2b=c\) và \(3a-c=2b\)

+ Tương tự ta cũng có \(3b-2c=a\) và \(3b-a=2c\)

Và \(3c-2a=b\); \(3c-b=2a\)

Thay vào P

\(P=\frac{c.a.b}{2.b.2.c.2.a}=\frac{1}{8}\)

\(\frac{3a+b+c}{a}=\frac{a+3b+c}{b}=\frac{a+b+3c}{c}=\frac{\left(3a+b+c\right)+\left(a+3b+c\right)+\left(a+b+3c\right)}{a+b+c}\)

\(=\frac{5\left(a+b+c\right)}{a+b+c}=5\)

\(\Rightarrow\frac{3a+b+c}{a}=5\Rightarrow3a+b+c=5a\Rightarrow b+c=2a\)

Tương tự ta có : \(a+c=2b;a+b=2c\)

\(\Rightarrow B=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)

\(=\frac{a+b}{b}.\frac{b+c}{c}.\frac{a+c}{a}=\frac{2c}{b}.\frac{2a}{c}.\frac{2b}{a}\)

\(=\frac{8abc}{abc}=8\)

Ta có:

\(\frac{3a+b+c}{a}=\frac{a+3b+c}{b}=\frac{3c+b+a}{c}\)

\(\Rightarrow3+\frac{b+c}{a}=3+\frac{a+c}{b}=3+\frac{b+a}{c}\)

\(\Rightarrow\frac{b+c}{a}=\frac{a+c}{b}=\frac{b+a}{c}=\frac{2\left(a+b+c\right)}{a+b+c}\)

TH1:\(a+b+c=0\)\(\Rightarrow\left\{{}\begin{matrix}a+b=-c\\b+c=-a\\a+c=-b\end{matrix}\right.\)

Thay vào B, ta có:

\(B=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=-1\)

TH2:\(a+b+c\ne0\)

\(\Rightarrow\frac{b+c}{a}=\frac{a+c}{b}=\frac{b+a}{c}=\frac{2\left(a+b+c\right)}{a+b+c}=2\)

\(\Rightarrow\left\{{}\begin{matrix}b+c=2a\\a+c=2b\\b+a=2c\end{matrix}\right.\)

Thay vào B, ta có:

\(B=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=8\)

Vậy \(\left[{}\begin{matrix}B=-1\\B=8\end{matrix}\right.\)

Giải toán trên mạng - Giúp tôi giải toán - Hỏi đáp, thảo luận về toán học - Học toán với OnlineMath

\(\frac{2a+a+b+c}{a}=\frac{2b+a+b+c}{b}=\frac{2c+a+b+c}{c}\)

\(\Rightarrow2+\frac{a+b+c}{a}=2+\frac{a+b+c}{b}=2+\frac{a+b+c}{c}\)

\(\Rightarrow\frac{a+b+c}{a}=\frac{a+b+c}{b}=\frac{a+b+c}{c}\Rightarrow a=b=c\)

\(\Rightarrow B=\left(1+1\right)\left(1+1\right)\left(1+1\right)\)

Sorry ko bt làm !

\(VP=\frac{6}{\sqrt{\left(3a+bc\right)\left(3b+ca\right)\left(3c+ab\right)}}\)

\(=\frac{6}{\sqrt{\left[\left(a+b+c\right)a+bc\right]\left[\left(a+b+c\right)b+ca\right]\left[\left(a+b+c\right)c+ab\right]}}\)

\(=\frac{6}{\sqrt{\left(a+b\right)^2\left(b+c\right)^2\left(c+1\right)^2}}=\frac{6}{\left(a+b\right)\left(b+c\right)\left(a+c\right)}\)

\(VT=\frac{1}{3a+bc}+\frac{1}{3b+ca}+\frac{1}{3c+ab}\)

\(=\frac{1}{\left(a+b+c\right)a+bc}+\frac{1}{\left(a+b+c\right)b+ac}+\frac{1}{\left(a+b+c\right)c+ab}\)

\(=\frac{\left(b+c\right)+\left(a+c\right)+\left(a+b\right)}{\left(a+b\right)\left(b+c\right)\left(a+c\right)}=\frac{6}{\left(a+b\right)\left(b+c\right)\left(a+c\right)}\)

Vậy VT = VP, đẳng thức được chứng minh