Problem:
Two parabolas have equations and , where , , and are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The parabolas have no points in common if and only if the equation has no solution. This is true if and only if the lines with equations and are parallel, which happens if and only if and . The probability that is and the probability that is , so the probability that the two parabolas have a point in common is .
The problems on this page are the property of the MAA's American Mathematics Competitions