Ta có:
1/(1+a)+1/(1+b)+1/(1+c)≥2
→1/(1+a)≥{1-1/(1+b)}+{1-1/(1+c)}
↔1/(1+a)≥b/(1+b)+c/(1+c)
≥2.√(bc)/{(1+b)(1+c)}(theo cosi)
Hai bất đẳng thức tương tự rồi nhân vế với vế
1/{(1+a)(1+b)(1+c)≥8.abc/{(1+a)(1+b)(1...
↔abc≤1/8(dpcm)
TK NHA
\(\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\ge2\Rightarrow\frac{1}{1+a}\ge\left(1-\frac{1}{1+b}\right)+\left(1-\frac{1}{1+c}\right)\)\(=\frac{b}{1+b}+\frac{c}{1+c}\ge2\sqrt{\frac{bc}{\left(1+b\right)\left(1+c\right)}}\)
Tương tự ta có:
\(\frac{1}{1+b}\ge2\sqrt{\frac{ac}{\left(1+a\right)\left(1+c\right)}};\frac{1}{1+c}\ge2\sqrt{\frac{ab}{\left(1+a\right)\left(1+b\right)}}\). Suy ra:
\(\frac{1}{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\ge\frac{8abc}{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\Rightarrow abc\le\frac{1}{8}.\)
