Problem: If a,b,c are non-zero real numbers such that
ca+b−c​=ba−b+c​=a−a+b+c​
and
x=abc(a+b)(b+c)(c+a)​​
and x<0, then x equals
Answer Choices:
A. −1
B. −2
C. −4
D. −6
E. −8
Solution:
Observe that
ca+b−c​+2=ba−b+c​+2=a−a+b+c​+2ca+b+c​=ba+b+c​=aa+b+c​​
These equalities are satisfied if a+b+c=0. If a+b+cî€ =0, then dividing each member of the second set of equalities by a+b+c and taking the reciprocals of the results yields a=b=c. If a+b+c=0, then x=−1; if a=b=c, then x=8.