Question

Cho a, b, c là các số thực thỏa mãn 1/a + 1/b +1/c = 1 và a + b + c = 1. Chứng minh rằng ( a-1)(b-1)(c-1) = 0

Answer 1

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-1=0\Leftrightarrow\frac{a+b}{ab}+\frac{1-c}{c}\Leftrightarrow\frac{1-c}{ab}+\frac{1-c}{c}=0\)

\(\Leftrightarrow\left(1-c\right)\left(\frac{1}{ab}+\frac{1}{c}\right)=0\Leftrightarrow\left(1-c\right)\left(\frac{c+ab}{abc}\right)=0\)

\(\Leftrightarrow\left(1-c\right)\left(\frac{1-a-b+ab}{abc}\right)=0\Leftrightarrow\frac{\left(1-c\right)\left(1-a\right)\left(1-b\right)}{abc}=0\)

\(\Leftrightarrow\left(1-c\right)\left(1-a\right)\left(1-b\right)=0\)