2020 AIME II Problems/Problem 8
Problem
Define a sequence recursively by and for integers . Find the least value of such that the sum of the zeros of exceeds .
Solution (Official MAA)
First it will be shown by induction that the zeros of are the integers , where
This is certainly true for . Suppose that it is true for , and note that the zeros of are the solutions of , where is a nonnegative zero of . Because the zeros of form an arithmetic sequence with common difference so do the zeros of . The greatest zero of isso the greatest zero of is and the least is .
It follows that the number of zeros of is , and their average value is . The sum of the zeros of isLet , so the sum of the zeros exceeds if and only if Because is increasing for , the values and show that the requested value of is .
Video Solution
~MathProblemSolvingSkills.com
Video Solution
~IceMatrix
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.