Question

帮忙解道题 ,已知x+1/y=y+1/z=z+1/x ,证明x+y+z=1

Answer 1

∵x+1/y=y+1/z ∴x-y=1/z-1/y=(y-z)/yz.(1) ∵y+1/z=z+1/x ∴y-z=1/x-1/z=(z-x)/xz.(2) ∵x+1/y=z+1/x ∴z-x=1/y-1/x=(x-y)/(xy).(3) （1）*（2）*（3）得：(x-y)(y-z)(z-x) = (x-y)(y-z)(z-x)/(x^2y^2z^2).(4) ∵x,y,z 为互不相等的非零实数 ∴（4）式两边除以(x-y)(y-z)(z-x) 得：1 = 1/(x^2y^2z^2) x^2y^2z^2 = 1