Question

P=(1+a/b)(1+b/c)(1+c/a) với a+b/c=b+c/a=a+c/b

Answer 1

Theo dãy tỉ số (=) ta có :

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

=> a+ b = 2c ; b+c = 2a ; a+ c = 2b

P  =\(\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=\frac{a+b}{b}.\frac{b+c}{c}+\frac{a+c}{c}=\frac{2c}{b}\cdot\frac{2a}{c}\cdot\frac{2b}{a}=\frac{8abc}{abc}=8\)

Answer 2

bấm vào đây hoặc cái này

Answer 3

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

=>a+b=2c;b+c=2a;a+c=2b

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

\(=\frac{2c}{b}.\frac{2a}{c}.\frac{2b}{a}=\frac{2.2.2\left(a.b.c\right)}{a.b.c}=8\)

vậy P=8