Question

已知abc满足a² +b²+ c²=1 a(1/b 1/c)+ b(1/a 1/c)+ c(1/a 1/b)=-3求a+ b +c

已知abc满足a² +b²+ c²=1 a(1/b 1/c) +b(1/a 1/c)+ c(1/a 1/b)=-3 求a+ b +c

Answer 1

a(1/b+1/c)+b(1/a+1/c)+c(1/a+1/b)+3
=(a + b + c) (a b + a c + b c)/(abc) =0

若 a b + a c + b c =0,则
(a+b+c)^2-1
=(a+b+c)^2 -(a^2+b^2+c^2)
=2(a b + a c + b c)=0
因此a+b+c 有三个值：0,-1,1