Problem:
Values for , and are to be selected from without replacement (i.e., no two letters have the same value). How many ways are there to make such choices so that the two curves and intersect? (The order in which the curves are listed does not matter; for example, the choices is considered the same as the choices .)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
All these parabolas open upward and are symmetric about the -axis. Because the selection of the coefficients for the two parabolas is made without replacement, the vertex and the narrowness of the first parabola are different from the vertex and the narrowness of the second parabola. The two parabolas intersect if and only if the vertex of the narrower parabola lies below the vertex of the wider parabola - the narrower one has the greater coefficient and the higher one is the one with the greater constant term. Therefore an intersection will occur if and only if and have opposite signs. There are choices for an unordered set of numbers from . Without loss of generality, let be the greater and the lesser, so that . There are choices for an unordered set of numbers from the remaining numbers. The parabolas will intersect if and only if is the lesser of the two, so that . Thus there are choices in all.
The problems on this page are the property of the MAA's American Mathematics Competitions