Question

cho \(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\)

chứng minh trong ba số a,b,c tồn tại hai số bằng nhau

Answer 1

\(abc\ne0\)

\(abc\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)=abc\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right)\)

\(\Leftrightarrow a^2c+ab^2+bc^2=b^2c+ac^2+a^2b\)

\(\Leftrightarrow a^2c-b^2c+ab^2-a^2b+bc^2-ac^2=0\)

\(\Leftrightarrow c\left(a-b\right)\left(a+b\right)-ab\left(a-b\right)-c^2\left(a-b\right)=0\)

\(\Leftrightarrow\left(a-b\right)\left(ac+bc-ab-c^2\right)=0\)

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

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

\(\Leftrightarrow\left[{}\begin{matrix}a=b\\a=c\\b=c\end{matrix}\right.\) (đpcm)