Question

cho a,b,c thỏa mãn

\(a^{2010}+b^{2010}+c^{2010}=a^{1005}b^{1005}+b^{1005}c^{1005}+c^{1005}a^{1005}\)

tính giá trị biểu thức \(M=\left(a-b\right)^{20}+\left(b-c\right)^{12}+\left(c-a\right)^{2013}\)

Answer 1

Đặt \(\left\{{}\begin{matrix}a^{1005}=x\\b^{1005}=y\\c^{1005}=z\end{matrix}\right.\) \(\Rightarrow x^2+y^2+z^2=xz+xz+yz\)

\(\Leftrightarrow2x^2+2y^2+2z^2=2xy+2xz+2yz\)

\(\Leftrightarrow x^2-2xy+y^2+x^2-2xz+z^2+y^2-2yz+z^2=0\)

\(\Leftrightarrow\left(x-y\right)^2+\left(x-z\right)^2+\left(y-z\right)^2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-y=0\\x-z=0\\y-z=0\end{matrix}\right.\) \(\Leftrightarrow x=y=z\)

\(\Rightarrow a^{1005}=b^{1005}=c^{1005}\Rightarrow a=b=c\)

\(\Rightarrow M=0\)

Answer 2

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Answer 3

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