Question

[Quang Huy Điền]

Cho a, b, c là 3 số khác nhau. CMR :

\(\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\)

Answer 1

[Rồng Đom Đóm]

Đặt \(A=\frac{b-c}{\left(a-b\right)\left(a-c\right)}+\frac{c-a}{\left(b-c\right)\left(b-a\right)}+\frac{a-b}{\left(c-a\right)\left(c-b\right)}\)

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

\(A=\frac{-b^2+2bc-c^2-a^2+2ac-c^2-a^2+2ab-b^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

\(\Rightarrow\frac{A}{2}=\frac{ab+bc+ca-a^2-b^2-c^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

Đặt \(B=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}\)

\(\frac{B}{2}=\frac{\left(b-c\right)\left(c-a\right)+\left(a-b\right)\left(c-a\right)+\left(a-b\right)\left(b-c\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

\(\frac{B}{2}=\frac{bc-ab-c^2+ac+ac-a^2-bc+ab+ab-ac-b^2+bc}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

\(\frac{B}{2}=\frac{ab+bc+ca-a^2-b^2-c^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)

\(\Rightarrow A=B\left(đpcm\right)\)