(a/b+c/b+ 1)+(b/a+c/a+ 1)+(a/c+b/c+ 1)= 0

(a+b+c)/b+(a+b+c)/a+(a+b+c)/c=0

(a+b+c)( 1/a+ 1/b+ 1/c)= 0

(a+b+c)(ab+bc+ca)/abc=0

So (a+b+c)(ab+bc+ca)=0.

Let k=a+b+c

k^2=a^2+b^2+c^2+2(ab+bc+ca)= 1+2(ab+bc+ca)

(ab+bc+ca)=(k^2- 1)/2

So k * (k 2-1)/2 = 0.

k=0,k^2- 1=0

k=0，k= 1，k=- 1

So a+b+c=0 or-1 or 1.