2021 AIME I Problems/Problem 15
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[hide]Problem
Let be the set of positive integers such that the two parabolasintersect 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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