一个简单的方法：
a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)
=a(1/a+1/b+1/c) + b(1/b+1/c+1/a) + c(1/c+1/a+1/b) -3
=(a+b+c)*(1/a+1/b+1/c)-3
=-3