Question

若abc不等于0，且a+b/c=b+c/a=a+c/b，求(a+b)(b+c)(c+a)/abc的值

Answer 1

令k=a+b/c=b+c/a=a+c/b
则a+b=ck
b+c=ak
a+c=bk
相加
2(a+b+c)=k(a+b+c)
(a+b+c)(k-2)=0

若a+b+c=0,则a+b=-c,b+c=-a,c+a=-b
则(a+b)(b+c)(c+a)/abc
=(-c)(-a)(-b)/abc
=-1

若k-2=0
k=2
则a+b=2c
b+c=2a
a+c=2b
则(a+b)(b+c)(c+a)/abc
=2c*2a*2b/abc
=8

所以则(a+b)(b+c)(c+a)/abc=-1或8

Answer 2

a+b/c=b+c/a=a+c/b=k
a+b=ck,b+c=ak,a+c=bk
2(a+b+c)=k(a+b+c)
k=2
(a+b)(b+c)(c+a)/abc
=k^3=8

Answer 3

1