Question

Cho ba số thực dương a,b,c thỏa mãn $ab\ge 12;bc\ge 8$. Chứng mình rằng:

\(\left(a+b+c\right)+2.\left(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ca}\right)+\dfrac{8}{abc}\ge\dfrac{121}{12}\)

 

Answer 1

Tách biểu thức như sau:

\(\left(\dfrac{a}{9}+\dfrac{b}{12}+\dfrac{c}{6}+\dfrac{8}{abc}\right)+\left(\dfrac{a}{18}+\dfrac{b}{24}+\dfrac{2}{ab}\right)+\left(\dfrac{b}{16}+\dfrac{c}{8}+\dfrac{2}{bc}\right)+\left(\dfrac{a}{9}+\dfrac{c}{6}+\dfrac{2}{ca}\right)+\left(\dfrac{13a}{18}+\dfrac{13b}{24}\right)+\left(\dfrac{13b}{48}+\dfrac{13c}{24}\right)\)

Answer 2

(Nháp)\(a+2b+3c=20\)Với các tham số \(0< x,y,z< 1\) ta có:\(A=a+b+c+\dfrac{3}{a}+\dfrac{9}{2b}+\dfrac{4}{c}\)\(=xa+yb+zc+\left(\dfrac{3}{a}+\left(1-x\right)a\right)+\left(\dfrac{9}{2b}+\left(1-y\right)b\right)+\left(\dfrac{4}{c}+\left(1-z\right)c\right)\)\(\ge^{Cauchy}xa+yb+zc+2\left(\sqrt{3\left(1-x\right)}+\sqrt{\dfrac{9\left(1-y\right)}{2}}+\sqrt{4\left(1-z\right)}\right)\)Chọn các tham số x,y,z (0<x,y,z<1) sao cho:\(\left\{{}\begin{matrix}x=\dfrac{y}{2}=\dfrac{z}{3}\\\dfrac{3}{a}=\left(1-x\right)a\\\dfrac{9}{2b}=\left(1-y\right)b\\\dfrac{4}{c}=\left(1-z\right)c\end{matrix}\right.\) và \(a+2b+3c=20\) \(\Rightarrow\left\{{}\begin{matrix}y=2x;z=3x\\a=\sqrt{\dfrac{3}{1-x}}\\b=\sqrt{\dfrac{9}{2\left(1-y\right)}}\\c=\sqrt{\dfrac{4}{1-z}}\end{matrix}\right.\) và \(a+2b+3c=20\)\(\Rightarrow\left\{{}\begin{matrix}y=2x;z=3x\\a=\sqrt{\dfrac{3}{1-x}}\\b=\sqrt{\dfrac{9}{2\left(1-2x\right)}}\\c=\sqrt{\dfrac{4}{1-3x}}\end{matrix}\right.\) và \(a+2b+3c=20\)\(\Rightarrow\sqrt{\dfrac{3}{1-x}}+2\sqrt{\dfrac{9}{2\left(1-2x\right)}}+3\sqrt{\dfrac{4}{1-3x}}=20\)Bấm máy ta được \(x=\dfrac{1}{4}\Rightarrow y=\dfrac{1}{2};z=\dfrac{3}{4}\)\(\Rightarrow\left\{{}\begin{matrix}a=\sqrt{\dfrac{3}{1-\dfrac{1}{4}}}=2\\b=\sqrt{\dfrac{9}{2\left(1-2.\dfrac{1}{4}\right)}}=3\\c=\sqrt{\dfrac{4}{1-3.\dfrac{1}{4}}}=4\end{matrix}\right.\)