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2022 Algebra/NT #5

Source:

March 11, 2022
algebra

Problem Statement

Given a positive integer kk, let ∣∣k∣∣||k|| denote the absolute difference between kk and the nearest perfect square. For example, ∣∣13∣∣=3||13||=3 since the nearest perfect square to 1313 is 1616. Compute the smallest positive integer nn such that ∣∣1∣∣+∣∣2∣∣+...+∣∣n∣∣n=100\frac{||1|| + ||2|| + ...+ ||n||}{n}=100.
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