2021 AIME I Problems/Problem 15

Revision as of 04:22, 12 March 2021 by Lopkiloinm (talk | contribs) (Solution)

Problem

Let $S$ be the set of positive integers $k$ such that the two parabolas\[y=x^2-k~~\text{and}~~x=2(y-20)^2-k\]intersect in four distinct points, and these four points lie on a circle with radius at most $21$. Find the sum of the least element of $S$ and the greatest element of $S$.

Solution

Using the computer science algorithm called binary search, you can narrow down the answer. Binary search takes 10 iterations because the range is 0 to 999 and log base 2 of 999 is 10.

See also

2021 AIME I (ProblemsAnswer KeyResources)
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