Question

[Yurii]

Cho a,b,c là ba số dương, thỏa mãn:

\(\frac{2016c-a-b}{c}=\)\(\frac{2016b-a-c}{b}=\)\(\frac{2016a-b-c}{a}\)

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

Answer 1

[Thanh Tùng DZ]

ADTCCDTSBN,TC :

\(\frac{2016c-a-b}{c}=\frac{2016b-a-c}{b}=\frac{2016a-b-c}{a}\)

\(=\frac{\left(2016c-a-b\right)+\left(2016b-a-c\right)+\left(2016a-b-c\right)}{c+b+a}=\frac{2014.\left(a+b+c\right)}{a+b+c}=2014\)

\(\frac{2016c-a-b}{c}=2014\Rightarrow2016c-a-b=2014c\Rightarrow2c=a+b\)( 1 )

\(\frac{2016b-a-c}{b}=2014\Rightarrow2016b-a-c=2014b\Rightarrow2b=a+c\)( 2 )

\(\frac{2016a-b-c}{a}=2014\Rightarrow2016a-b-c=2014a\Rightarrow2a=b+c\)( 3 )

Từ ( 1 ), ( 2 ) và ( 3 ) \(\Rightarrow\)a = b = c

\(\Rightarrow A=\left(1+\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)=\left(1+1\right)\left(1+1\right)+\left(1+1\right)=2^3=8\)