Problem:
What is the sum of the reciprocals of the roots of the equation
20042003x+1+x1=0?
Answer Choices:
A. −20032004
B. −1
C. 20042003
D. 1
E. 20032004
Solution:
Let a=2003/2004. The given equation is equivalent to
ax2+x+1=0.
If the roots of this equation are denoted r and s, then
rs=a1 and r+s=−a1,
so
r1+s1=rsr+s=−1.
OR
If x is replaced by 1/y, then the roots of the resulting equation are the reciprocals of the roots of the original equation. The new equation is
2004y2003+1+y=0 which is equivalent to y2+y+20042003=0.
The sum of the roots of this equation is the opposite of the y-coefficient, which is −1.
Answer: B.
The problems on this page are the property of the MAA's American Mathematics Competitions