Question

Cho abc≠ \(\pm1\) và \(\dfrac{ab+1}{b}=\dfrac{bc+1}{c}=\dfrac{ca+1}{a}\). Chứng minh rằng a=b=c

Answer 1

(a,b,c khác 0 nữa)

\(\dfrac{ab+1}{b}=\dfrac{bc+1}{c}=\dfrac{ca+1}{a}\)

\(\Leftrightarrow a+\dfrac{1}{b}=b+\dfrac{1}{c}=c+\dfrac{1}{a}\)

\(\Leftrightarrow\left\{{}\begin{matrix}a-b=\dfrac{c-b}{bc}\\b-c=\dfrac{a-c}{ca}\\c-a=\dfrac{b-a}{ab}\end{matrix}\right.\)(1)

Xét a=b hoặc b=c hoặc c=a thì=>a=b=c

Xét \(a\ne b\ne c\)

\(\left(1\right)\Leftrightarrow\left(a-b\right)\left(b-c\right)\left(c-a\right)=\dfrac{\left(c-b\right)\left(a-c\right)\left(b-a\right)}{a^2b^2c^2}\)

\(\Leftrightarrow-1=\dfrac{1}{a^2b^2c^2}\)(vô nghiệm)

Vậy ...