Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 6"
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*[[Mock AIME 1 2006-2007]] | *[[Mock AIME 1 2006-2007]] |
Latest revision as of 15:52, 3 April 2012
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
.