Question

Cho a> 0, b>0, c>0 . CMR: a² / ( a +b) + b² / ( b +c ) + c²/ ( c +a ) >= ( a +b +c )/2

Answer 1

Áp dụng BĐT Cauchy-Schwarz dạng engel:

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

Dấu "=" xảy ra khi \(a=b=c\)

Answer 2

Cách khác :

Áp dụng BĐT AM-GM cho 2 số dương ta có:

\(\dfrac{a^2}{a+b}+\dfrac{a+b}{4}\ge2\sqrt{\dfrac{a^2\left(a+b\right)}{4\left(a+b\right)}}=a\)

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

Cộng theo vế ta được:

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

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