Question

Cho số thực a,b,x,y thỏa mãn a+b=x+y và a²+b²=x²+y²

CMR:a²⁰¹⁷+b²⁰¹⁷=x²⁰¹⁷+y²⁰¹⁷

Answer 1

\(a+b=x+y\Rightarrow a^2+2ab+b^2=x^2+2xy+y^2\Rightarrow2ab=2xy\Rightarrow a^2-2ab+b^2=x^2-2xy+y^2\Rightarrow\left(a-b\right)^2=\left(x-y\right)^2\Rightarrow\left[{}\begin{matrix}a-b=x-y\\a-b=y-x\end{matrix}\right.\) \(+,a-b=x-y\Rightarrow a+b+\left(a-b\right)=x+y+\left(x-y\right)\Rightarrow2a=2x\Rightarrow a=x\Rightarrow b=y\Rightarrow a^{2017}+b^{2017}=x^{2017}+y^{2017}\) \(+,a-b=y-x\Rightarrow\left(a+b\right)+\left(a-b\right)=x+y+\left(y-x\right)\Rightarrow2a=2y\Rightarrow a=y\Rightarrow b=x\Rightarrow a^{2017}+b^{2017}=x^{2017}+y^{2017}\)

\(\Rightarrow dpcm\)