2021 AIME I Problems/Problem 15
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 .
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Solution 1 (Inequalities and Circles)
Note that is an upward-opening parabola with the vertex at and is a rightward-opening parabola with the vertex at We consider each condition separately:
- The two parabolas intersect at four distinct points.
- The point is on or below the parabola
We need from which
Moreover, the point is on the parabola when We will prove that the two parabolas intersect at four distinct points at this value of
Substituting into we get Expanding and rearranging give By either the graphs of the parabolas or the Rational Root Theorem, we conclude that is a root of So, we factor its left side: By either the graphs of the parabolas or Descartes' Rule of Signs, we conclude that has two positive roots and one negative root such that So, has four distinct real roots, or the two parabolas intersect at four distinct points.
For Subcondition A, we deduce that
Remark for Subcondition A
Recall that if then the point is above the parabola It follows that in the interval
- The maximum value of for the parabola occurs at from which
- The minimum value of for the parabola occurs at from which
Clearly, the parabola and the left half of the parabola do not intersect. Therefore, the two parabolas do not intersect at four points.
- The point is on or below the parabola
The lower half of the parabola is We need which holds for all values of
For Subcondition B, we deduce that can be any positive integer.
- The four points of intersection lie on a circle with radius at most
For equations of circles, the coefficients of and must be the same. So, we add the equation to half the equation We expand, rearrange, and complete the squares: We need from which
For Condition 2, we obtain
By a quick sketch, we have two subconditions:
For Condition 1, we obtain by taking the intersection of Subconditions A and B.
Taking the intersection of Conditions 1 and 2 produces Therefore, the answer is
Solution 2 (Translations, Inequalities, Circles)
Make the translation to obtain and . Multiply the first equation by 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 intersections to begin with. (Here I'm using the un-translated coordinates.) Draw up a graph, and realize that two intersections are guaranteed, on the so called "right branch" of . As we increase the value of , two more intersections appear on the "left branch":
does not work because the "leftmost" point of is which lies to the right of , which is on the graph . While technically speaking this doesn't prove that there are no intersections (why?), drawing the graph should convince you that this is the case. Clearly, does not work.
does work because the two graphs intersect at , and by drawing the graph, you realize this is not a tangent point and there is in fact another intersection nearby, due to slope. Therefore, the answer is .
- In general (assuming four intersections exist), when two conics intersect, if one conic can be written as and the other as for polynomials and of degree at most , whenever are linearly independent (L.I.), 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 or intersection point(s), the statement that all these points lie on a circle is trivially true.
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