Question

cho a,b,c dương thỏa mãn\(\frac{ab}{a+b}=\frac{bc}{b+c}=\frac{ac}{a+c}\)tính giá trị \(A=\frac{a^3+b^3+c^3}{a^2b+b^2c+c^2a}\)

Answer 1

\(\frac{ab}{a+b}=\frac{bc}{b+c}\Rightarrow\frac{a}{a+b}=\frac{c}{b+c}\Rightarrow a\left(b+c\right)=c\left(a+b\right)\)

\(\Rightarrow ab+ac=ac+bc\Rightarrow a=c\)

\(\frac{bc}{b+c}=\frac{ac}{a+c}\Rightarrow\frac{b}{b+c}=\frac{a}{a+c}\Rightarrow b\left(a+c\right)=a\left(b+c\right)\)

\(\Leftrightarrow ab+bc=ab+ac\Rightarrow a=b\)

=> a=b=c

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

Answer 2

\(A=4\)