2021 AIME I Problems/Problem 15
Contents
Problem
Let be the set of positive integers
such that the two parabolas
intersect in four distinct points, and these four points lie on a circle with radius at most
. Find the sum of the least element of
and the greatest element of
.
Solution 1
Make the translation to obtain
. Multiply the first equation by 2 and sum, we see that
. Completing the square gives us
; this explains why the two parabolas intersect at four points that lie on a circle*. For the upper bound, observe that
, so
.
For the lower bound, we need to ensure there are 4 intersections to begin with. A quick check shows k=5 works while k=4 does not. Therefore, the answer is 5+280=285.
- In general, (Assuming four intersections exist) when two conics intersect, if one conic can be written as
and the other as
for f,g polynomials of degree at most 1, whenever
are linearly independent, we can combine the two equations and then complete the square to achieve
. We can also combine these two equations to form a parabola, or a hyperbola, or an ellipse. When
are not L.I., the intersection points instead lie on a line, which is a circle of radius infinity. When the two conics only have 3,2 or 1 intersection points, the statement that all these points lie on a circle is trivially true.
-Ross Gao
Solution 2
Let is first parabola with axis
and vertex at
and
be the second parabola with axis at
with vertex at
.
Vertex for the
vertex is at
, so
is intersecting
only at two points. As we increase
,
's vertex gets closer to
. It intersects
at
.
Plugging in in
.
Note gives exactly
intersections between
and
. For
and
to have 4 intersections, the smallest
needs to be
and corresponding circle will be the smallest circle.
We do realize that as increases beyond
the number of intersections remain
but the radius of the common intersecting circle will increase. Consider the largest circle of radius
and test if an integer
that satisfies the common intersection between
.
The circle will be symmetric about y-axis and line with center at
. So the general equation of circle
Using
and
we get a line equation
Solving for using largest circle
and
:
Solving
we get:
Solving
we get:
Plugging for pairs in
we get
=
; the value of
satisfies (1) is
meaning
Hence
~Math_Genius_164
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
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