Question

[Khánh Ngọc]

1. Cho 3 số dương a, b, c thỏa mãn ab + bc + ca = 3abc 

Tính GTNN của bt : \(M=\frac{2\left(a^2b^2+b^2c^2+c^2a^2\right)+abc}{a^2b^2c^2}\)

2. Cho a, b, c\(\inℝ^+\)thỏa mãn a + b + c = 4. Cmr BĐT sau luôn đúng :

\(10\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right)\ge\frac{4+5a}{4-a}+\frac{4+5b}{4-b}+\frac{4+5c}{4-c}\)

Answer 1

[Nguyễn Minh Đăng]

1. Ta có: \(ab+bc+ca=3abc\)

\(\Leftrightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3\)

Đặt \(\hept{\begin{cases}\frac{1}{a}=m\\\frac{1}{b}=n\\\frac{1}{c}=p\end{cases}}\) khi đó \(\hept{\begin{cases}m+n+p=3\\M=2\left(m^2+n^2+p^2\right)+mnp\end{cases}}\)

Áp dụng Cauchy ta được:

\(\left(m+n-p\right)\left(m-n+p\right)\le\left(\frac{m+n-p+m-n+p}{2}\right)^2=m^2\)

\(\left(n+p-m\right)\left(n+m-p\right)\le n^2\)

\(\left(p-n+m\right)\left(p-m+n\right)\le p^2\)

\(\Rightarrow\left(m+n-p\right)\left(n+p-m\right)\left(p+m-n\right)\le mnp\)

\(\Leftrightarrow m^3+n^3+p^3+3mnp\ge m^2n+mn^2+n^2p+np^2+p^2m+pm^2\)

\(\Leftrightarrow\left(m+n+p\right)\left(m^2+n^2+p^2-mn-np-pm\right)+6mnp\ge mn\left(m-n\right)+np\left(n-p\right)+pm\left(p-m\right)\)

\(=mn\left(3-p\right)+np\left(3-m\right)+pm\left(3-n\right)\)

\(\Leftrightarrow3\left(m^2+n^2+p^2\right)-3\left(mn+np+pm\right)+6mnp\ge3\left(mn+np+pm\right)-3mnp\)

\(\Leftrightarrow3\left(m^2+n^2+p^2\right)+9mnp\ge6\left(mn+np+pm\right)\)

\(\Leftrightarrow xyz\ge\frac{2}{3}\left(mn+np+pm\right)-\frac{1}{3}\left(m^2+n^2+p^2\right)\)

\(\Rightarrow M\ge2\left(m^2+n^2+p^2\right)+\frac{2}{3}\left(mn+np+pm\right)-\frac{1}{3}\left(m^2+n^2+p^2\right)\)

\(=\frac{5}{3}\left(m^2+n^2+p^2\right)+\frac{2}{3}\left(mn+np+pm\right)\)

\(=\frac{4}{3}\left(m^2+n^2+p^2\right)+\frac{1}{3}\left(m^2+n^2+p^2+2mn+2np+2pm\right)\)

\(=\frac{4}{3}\left(m^2+n^2+p^2\right)+\frac{1}{3}\left(m+n+p\right)^2\)

\(\ge\frac{4}{3}\cdot3+\frac{1}{3}\cdot3^2=4+3=7\)

Dấu "=" xảy ra khi: \(m=n=p=1\Leftrightarrow a=b=c=1\)