Problem: If a,b,c are non-zero real numbers such that
ca+bβcβ=baβb+cβ=aβa+b+cβ
and
x=abc(a+b)(b+c)(c+a)ββ
and x<0, then x equals
Answer Choices:
A. β1
B. β2
C. β4
D. β6
E. β8
Solution:
Observe that
ca+bβcβ+2=baβb+cβ+2=aβa+b+cβ+2ca+b+cβ=ba+b+cβ=aa+b+cββ
These equalities are satisfied if a+b+c=0. If a+b+cξ =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=β1; if a=b=c, then x=8.