Question

Cho x⁶⁷¹+y⁶⁷¹ =1 và x¹³⁴² + y¹³⁴² =2. Tính x²⁰¹³ + y²⁰¹³

Answer 1

\(x^{671}+y^{671}=1\Rightarrow\left(x^{671}+y^{671}\right)^2=x^{1342}+2.x^{671}.y^{671}+y^{1342}\)\(=1\)

Mà \(x^{1342}+y^{1342}=2\) \(\Rightarrow x^{671}.y^{671}=\dfrac{-1}{2}\)

Mặt khác: \(\left(x^{671}+y^{671}\right)^3=x^{2013}+3x^{671}y^{671}\left(x^{671}+y^{671}\right)+y^{2013}=1\)

Hay \(x^{2013}+y^{2013}-\dfrac{3}{2}.1=1\Rightarrow x^{2013}+y^{2013}=1+\dfrac{3}{2}=\dfrac{5}{2}\)