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A set with fixed number of elements

Source: Romanian District Olympiad 2014, Grade 10, P3

June 15, 2014
floor functionlogarithmsalgebra proposedalgebra

Problem Statement

Let pp and nn be positive integers, with p≥2p\geq2, and let aa be a real number such that 1≤a<a+n≤p1\leq a<a+n\leq p. Prove that the set S={⌊log⁡2x⌋+⌊log⁡3x⌋+⋯+⌊log⁡px⌋∣x∈R,a≤x≤a+n} \mathcal {S}=\left\{\left\lfloor \log_{2}x\right\rfloor +\left\lfloor \log_{3}x\right\rfloor +\cdots+\left\lfloor \log_{p}x\right\rfloor\mid x\in\mathbb{R},a\leq x\leq a+n\right\} has exactly n+1n+1 elements.
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