Question

Cho a+b+c=2016

CMR  (a^3 +b^3 +c^3 -3abc) /(a^2 +b^2+c^2 -ab -ac -bc)=2016

 

Answer 1

Xét : \(a^3+b^3+c^3=\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc\)

\(=\left(a+b+c\right)\left(a^2+b^2+c^2+2ab-bc-ac\right)-3ab\left(a+b+c\right)\)

\(=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)\)

Suy ra : \(\frac{a^3+b^3+c^3-3abc}{a^2+b^2+c^2-ab-bc-ac}=\frac{\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)}{a^2+b^2+c^2-ab-bc-ac}=a+b+c=2016\)

Vậy ta có điều phải chứng minh.