已知x+y=a+b,x^2+y^2=a^2+b^2,求证x^2011+y^2011=a^2011+b^2011

Question

已知x+y=a+b,x^2+y^2=a^2+b^2,求证x^2011+y^2011=a^2011+b^2011

Answer 1

x+y=a+b (1)
(x+y)^2=(a+b)^2 x^2+2xy+y^2=a^2+2ab+b^2 (2)
x^2+y^2=a^2+b^2 (3)
(2)-(3) 2xy=2ab (4)
(3)-(4) (x-y)^2=(a-b)^2
x-y=±(a-b)
(a) x-y=a-b 与(1)联立解得 x=a y=b
则x^2011+y^2011=a^2011+b^2011 得证
(b) x-y=b-a 与(1)联立解得 x=b y=a
则同样x^2011+y^2011=a^2011+b^2011 得证
综上,x^2011+y^2011=a^2011+b^2011
始终成立.