Problem:
Let a and b be the roots of the equation x2−mx+2=0. Suppose that a+(1/b) and b+(1/a) are the roots of the equation
x2−px+q=0.
What is q?
Answer Choices:
A. 25
B. 27
C. 4
D. 29
E. 8
Solution:
Since a and b are roots of x2−mx+2=0, we have
x2−mx+2=(x−a)(x−b) and ab=2
In a similar manner, the constant term of x2−px+q is the product of a+(1/b) and b+(1/a), so
q=(a+b1)(b+a1)=ab+1+1+ab1=(D)29.
The problems on this page are the property of the MAA's American Mathematics Competitions