Question

[Lê Hương Giang]

Cho (a + b + c)(ab +bc + ca) = abc. Chứng minh rằng

a²⁰²¹ + b²⁰²¹ + c²⁰²¹ = (a + b + c)²⁰²¹

Answer 1

[Nguyễn Việt Lâm]

\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc=0\)

\(\Leftrightarrow\left(a+b\right)\left(ab+bc+ca\right)+c\left(ab+bc+ca\right)-abc=0\)

\(\Leftrightarrow\left(a+b\right)\left(ab+bc+ca\right)+c\left(bc+ca\right)=0\)

\(\Leftrightarrow\left(a+b\right)\left(ab+bc+ca+c^2\right)=0\)

\(\Leftrightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)=0\Rightarrow\left[{}\begin{matrix}a=-b\\b=-c\\c=-a\end{matrix}\right.\)

- Với \(a=-b\Rightarrow a^{2021}=-b^{2021}\Rightarrow\left\{{}\begin{matrix}a^{2021}+b^{2021}+c^{2021}=c^{2021}\\\left(a+b+c\right)^{2021}=c^{2021}\end{matrix}\right.\)

\(\Rightarrow a^{2021}+b^{2021}+c^{2021}=\left(a+b+c\right)^{2021}\)

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