2017 AIME I Problems/Problem 13
Problem 13
For every , let be the least positive integer with the following property: For every , there is always a perfect cube in the range . Find the remainder when is divided by 1000.
Solution
Lemma 1: The ratios between and decreases as increases.
Lemma 2: If the range includes two cubes, will always contain at least one cube for all integers in .
If , the range includes one cube. The range includes 2 cubes, which fulfills the Lemma. Since also included a cube, we can assume that for all . Two groups of 1000 are included in the sum modulo 1000. They do not count since for all of them, therefore
See Also
2017 AIME I (Problems • Answer Key • Resources) | ||
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Followed by Problem 14 | |
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