a)Cho a,b,c là các số thực dương thỏa mãn \(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ca}\) ≤ 1
Chứng minh rằng \(\dfrac{1}{a^2\left(1+bc\right)}+\dfrac{1}{b^2\left(1+ca\right)}+\dfrac{1}{c^2\left(1+ab\right)}\) ≤ \(\dfrac{1}{4}\).
b) Cho a,b,c là các số dương thỏa mãn \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\) = 4 Chứng minh rằng :
\(\dfrac{1}{2a+b+c}+\dfrac{1}{a+2b+c}+\dfrac{1}{a+b+2c}\) ≤ 1
b) \(\dfrac{1}{2a+b+c}+\dfrac{1}{2b+c+a}+\dfrac{1}{2c+a+b}\)
\(=\dfrac{1}{a+a+b+c}+\dfrac{1}{b+b+c+a}+\dfrac{1}{c+c+a+b}\)
\(\le\dfrac{1}{4\sqrt[4]{a.a.b.c}}+\dfrac{1}{4\sqrt[4]{b.b.c.a}}+\dfrac{1}{4\sqrt[4]{c.c.a.b}}\)
\(=\dfrac{1}{4}.\sqrt[4]{\dfrac{1}{a}.\dfrac{1}{a}.\dfrac{1}{b}.\dfrac{1}{c}}+\dfrac{1}{4}.\sqrt[4]{\dfrac{1}{b}.\dfrac{1}{b}.\dfrac{1}{c}.\dfrac{1}{a}}+\dfrac{1}{4}.\sqrt[4]{\dfrac{1}{c}.\dfrac{1}{c}.\dfrac{1}{a}.\dfrac{1}{b}}\)
\(\le\dfrac{1}{4}.\left(\dfrac{\dfrac{1}{a}+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}}{4}+\dfrac{\dfrac{1}{b}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a}}{4}+\dfrac{\dfrac{1}{c}+\dfrac{1}{c}+\dfrac{1}{a}+\dfrac{1}{b}}{4}\right)\)
\(=\dfrac{1}{4}\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\le\dfrac{1}{4}.4=1\left(đpcm\right)\)
- Dấu "=" xảy ra khi \(a=b=c=\dfrac{3}{4}\)
