Question

[Nguyen Phu Tho]

cho a,b,c # doi 1 va a+b/c=b+c/a=c+a/b. tinh

M=(1+ a/b)(1+ b/c)(1+ c/a)

Answer 1

[Pham Van Hung]

\(\frac{a+b}{c}=\frac{b+c}{a}=\frac{c+a}{b}\Rightarrow\frac{a+b+c}{c}=\frac{a+b+c}{a}=\frac{a+b+c}{b}\)    (cộng 3 vế với 1)

TH1: \(a+b+c=0\)

Khi đó: \(M=\left(\frac{a+b}{b}\right)\left(\frac{b+c}{c}\right)\left(\frac{c+a}{a}\right)=\frac{-c}{b}.\frac{-a}{c}.\frac{-b}{a}=-1\)

TH2: \(a=b=c\) (ko thỏa mãn a,b,c đôi 1 khác nhau)

Vây M = -1

Chúc bạn học tốt.

Answer 2

[I don't need you]

ta có: \(\frac{a+b}{c}=\frac{b+c}{a}=\frac{c+a}{b}=\frac{a+b+b+c+c+a}{c+a+b}=\frac{2.\left(a+b+c\right)}{a+b+c}.\)

Nếu \(a+b+c\ne0\)thì \(\frac{2.\left(a+b+c\right)}{a+b+c}=2\)

=> a + b = 2c

b+c = 2a

=> a-c = 2.(c-a)

=> c=a ( trái với đề bài)

=> a + b +c = 0

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