2017 AIME I Problems/Problem 13

Problem 13

For every $m \geq 2$ , let $Q(m)$ be the least positive integer with the following property: For every $n \geq Q(m)$ , there is always a perfect cube $k^3$ in the range $n < k^3 \leq m \cdot n$ . Find the remainder when $\sum_{m = 2}^{2017} Q(m)$ is divided by 1000.

Solution

Lemma: The ratios between $k^3$ and $(k+1)^3$ decreases as $k$ increases.

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2017 AIME I Problems/Problem 13

Problem 13

Solution