Question

\(\frac{1}{a+3b}+\frac{1}{b+3c}+\frac{1}{c+3a}\ge\frac{1}{a+b+2c}+\frac{1}{b+c+2a}+\frac{1}{a+c+2b}\)

Answer 1

Áp dụng BĐT \(\dfrac{1}{x}+\dfrac{1}{y}\ge\dfrac{4}{x+y}\) ta có:

\(\dfrac{1}{a+3b}+\dfrac{1}{a+b+2c}\ge\dfrac{4}{2a+4b+2c}=\dfrac{2}{a+2b+c}\)

Tương tự cho 2 BĐT còn lại ta cũng có:

\(\dfrac{1}{b+3c}+\dfrac{1}{2a+b+c}\ge\dfrac{2}{a+b+2c};\dfrac{1}{c+3a}+\dfrac{1}{a+2b+c}\ge\dfrac{2}{2a+b+c}\)

Cộng theo vế 3 BĐT trên ta có:

\(VT=\dfrac{1}{b+3c}+\dfrac{1}{c+3a}+\dfrac{1}{a+3b}\)

\(\ge\dfrac{1}{a+b+2c}+\dfrac{1}{2a+b+c}+\dfrac{1}{a+2b+c}=VP\)