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Identity with kth roots and logarithms

Source: Romania TST 2 2012, Problem 1

May 10, 2012
logarithmsfloor functionfunctioninductionalgebra proposedalgebra

Problem Statement

Prove that for any positive integer n≥2n\geq 2 we have that ∑k=2n⌊nk⌋=∑k=2n⌊log⁡kn⌋.\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.
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