Question

cho a,b,c>0 CMR:(a²/2b+3c)+(b^2/2c+3a)+(c²/2a+3b)<=1/8(a+b+c)

Answer 1

Sửa đề: CMR: \(\frac{a^2}{2a+3b}+\frac{b^2}{2b+3c}+\frac{c^2}{2c+3a}\ge\frac{1}{5}\left(a+b+c\right)\)

Chứng minh BĐT phụ:

  \(\frac{x^2}{m}+\frac{y^2}{n}\ge\frac{\left(x+y\right)^2}{m+n}\)\(\forall m;n>0\)Tự chứng minh

Áp dụng bđt trên, ta có

\(\frac{a^2}{2a+3b}+\frac{b^2}{2b+3c}+\frac{c^2}{2c+3a}\ge\frac{\left(a+b+c\right)^2}{2a+3b+2b+3c+2c+3a}=\frac{1}{5}\left(a+b+c\right)\)

Vậy..........