# Mock AIME 1 2006-2007 Problems/Problem 6

## Problem

Let and be two parabolas in the Cartesian plane. Let be the common tangent line of and that has a rational slope. If is written in the form for positive integers where , find .

## Solution

From the condition that is tangent to we have that the system of equations and has exactly one solution, so has exactly one solution. A quadratic equation with only one solution must have discriminant equal to zero, so we must have or equivalently . Applying the same process to , we have that has a unique root so or equivalently . We multiply the first of these equations through by and the second through by and subtract in order to eliminate and get . We know that the slope of , , is a rational number, so we divide this equation through by and let to get . Since we're searching for a rational root, we can use the Rational Root Theorem to search all possibilities and find that is a solution. (The other two roots are the roots of the quadratic equation , both of which are irrational.) Thus . Now we go back to one of our first equations, say , to get . (We can reject the alternate possibility because that would give and our "line" would not exist.) Then and since the greatest common divisor of the three numbers is 1, and .