Question

cmr:(a2/b+c)+(b2/a+c)+(c2/a+b)>a+b+c/2

Answer 1

Có điều kiện gì của a,b,c không ạ?

Answer 2

\(\dfrac{a^2}{b+c}+\dfrac{b^2}{a+c}+\dfrac{c^2}{a+b}\) (a,b,c thực dương)

=\(\dfrac{a^2}{b+c}+\dfrac{b+c}{4}+\dfrac{b^2}{a+c}+\dfrac{a+c}{4}+\dfrac{c^2}{a+b}+\dfrac{a+b}{4}\)

\(-\left(\dfrac{b+c}{4}+\dfrac{a+c}{4}+\dfrac{a+b}{4}\right)\)

áp dụng BDT Cô si =>\(\dfrac{a^2}{b+c}+\dfrac{b+c}{4}\ge a\)

tương tự : \(\dfrac{b^2}{a+c}+\dfrac{a+c}{4}\ge b\)

\(\dfrac{c^2}{a+b}+\dfrac{a+b}{4}\ge c\)

=>\(\dfrac{a^2}{b+c}+\dfrac{b+c}{4}+\dfrac{b^2}{a+c}+\dfrac{a+c}{4}+\dfrac{c^2}{a+b}+\dfrac{a+b}{4}\)

-\(-\left(\dfrac{b+c}{4}+\dfrac{a+c}{4}+\dfrac{a+b}{4}\right)\ge a+b+c-\dfrac{a+b+c}{2}\)

=\(\dfrac{a+b+c}{2}\left(dpcm\right)\)