已知非零实数a,b,c满足a^+b^+c^=1,且a(1/b+1/c）+b(1/c+1/a)+c(1/a+1/b）=-3，求a+b+c的值?

a(1/b+1/c）+b(1/c+1/a)+c(1/a+1/b）=-3
=>a(1/b+1/c）+b(1/c+1/a)+c(1/a+1/b）+3=0
=>a/b+a/c+b/b+b/a+c/a+c/b+3=0
=>(a+b+c)/b+(a+b+c)/a+(a+b+c)/c=0
=>(a+b+c)(ab+bc+ca)/abc=0
则因a,b,c为非0实数,a+b+c=0,or,ab+bc+ca=0
若ab+bc+ca=0
(a+b+c)²=a²+b²+c²+2ab+2bc+2ca=1
则a+b+c=0,±1

a(1/b+1/c）+b(1/c+1/a)+c(1/a+1/b）=-3
a(1/b+1/c）+b(1/c+1/a)+c(1/a+1/b）+3=0
a(1/b+1/c）+b(1/c+1/a)+c(1/a+1/b）+a/a+b/b+c/c=0
(a+b+c)/b+(a+b+c)/a+(a+b+c)/c=0
(a+b+c)(1/a+1/b+1/c)=0
所以a+b+c=0或者1/a+1/b+1/c=0
当1/a+1/b+1/c=0时，即ab+bc+ca=0 时
则（a+b+c)^=a^+b^+c^+2ab+2bc+2ca=1，
所以a+b+c=1或者a+b+c=-1，
综上所述，a+b+c的值为 0或者1或者-1.