Question

Cho a, b, c biết\(\frac{a+b}{c}=\frac{b+c}{a}=\frac{a+c}{b}\)

Tính  \(M=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\)

Answer 1

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

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

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

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

\(\Rightarrow a=b=c\) 

\(\Rightarrow\frac{a}{b}=1;\frac{b}{c}=1;\frac{c}{a}=1\)

\(\Rightarrow M=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=\left(1+1\right)\left(1+1\right)\left(1+1\right)=8\)