Question

Cho a,b,c thuộc R CMR \(\frac{a^2}{a^2+2bc}+\frac{b^2}{b^2+2ac}+\frac{c^2}{c^2+2ab}\ge1\)

Answer 1

Áp dụng bđt Cauchy Schwarz dạng Engel ta được:

\(\frac{a^2}{a^2+2bc}+\frac{b^2}{b^2+2ac}+\frac{c^2}{c^2+2ab}\ge\frac{\left(a+b+c\right)^2}{a^2+2bc+b^2+2ac+c^2+2ab}=\frac{\left(a+b+c\right)^2}{\left(a+b+c\right)^2}\)=1

Answer 2

Áp dụng bđt Cauchy-Schwarz dạng Engel ta có :

\(VT\ge\frac{\left(a+b+c\right)^2}{a^2+2bc+b^2+2ac+c^2+2ab}=\frac{\left(a+b+c\right)^2}{\left(a+b+c\right)^2}=1\)

=> đpcm

Dấu "=" xảy ra <=> a = b = c