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Sums of floor functions

Source: 44th International Tournament of Towns, Senior O-Level P2 and Junior O-Level P4, Spring 2023

January 9, 2024
floor functionnumber theory

Problem Statement

А positive integer nn{} is given. For every xx{} consider the sum Q(x)=∑k=110n⌊xk⌋.Q(x)=\sum_{k=1}^{10^n}\left\lfloor\frac{x}{k}\right\rfloor.Find the difference Q(10n)−Q(10n−1)Q(10^n)-Q(10^n-1).
Alexey Tolpygo
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