Question

CMR nếu: a^3+b^3+c^3=3abc và a,b,c là các số dương thì a=b=c.

Answer 1

\(a^3+b^3+c^3=3abc\Rightarrow a^3+b^3+c^3-3abc=0\)

\(\Rightarrow\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc=0\Rightarrow\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc=0\)

\(\Rightarrow\left(a+b+c\right)\left(\left(a+b\right)^2-c\left(a+b\right)+c^2\right)-3ab\left(a+b+c\right)=0\)

\(\Rightarrow\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2-3ab\right)=0\)

\(\left(a+b+c\right)\left(a^2-ab+b^2-bc+c^2-ca\right)=0\)\(Màa,b,c\ne0\Rightarrow a^2-ab+b^2-bc+c^2-ca=0\Rightarrow a\left(a-b\right)+b\left(b-c\right)+c\left(c-a\right)=0\)

\(a,b,c\ne0\Rightarrow a-b=0;b-c=0;c-a=0\Rightarrow a=b=c\)