2021 AIME I Problems/Problem 15
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
Solution 1
With binary search you can narrow down the k value. Newton raphson method let you narrow down the x and y solution for that specific k value. With 3 (x,y) pairs you can find radius of the circle.
You end up finding the bounds of 5 and 280. The sum is 285.
~Lopkiloinm
Solution 2
Make the translation to obtain
. Multiply the second 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, this problem tells us that the intersection points of two conics without xy terms usually lie on a circle. When is this true/false will be left to the reader.
-Ross Gao
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
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