Question

Cho tích a*b*c=1 và a+b+c > 1/a +1/b+1/c. Chứng minh rằng (a-1)*(b-1)*(c-1)>0

 

Answer 1

\(a+b+c>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)

\(\Leftrightarrow a+b+c>\frac{bc+ac+ab}{abc}\)

\(\Leftrightarrow a+b+c>bc+ac+ab\)

\(\Leftrightarrow a+b+c-bc-ac-ab>0\)

\(\Leftrightarrow abc+a+b+c-bc-ac-ab-abc>0\)

\(\Leftrightarrow abc+a+b+c-bc-ac-ab-1>0\)

\(\Leftrightarrow ab\left(c-1\right)-a\left(c-1\right)-b\left(c-1\right)+\left(c-1\right)>0\)

\(\Leftrightarrow\left(ab-a-b+1\right)\left(c-1\right)>0\)

\(\Rightarrow\left(a-1\right)\left(b-1\right)\left(c-1\right)>0\) (đpcm)