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\sum_{p\in P}\frac{1}{\log_p a_p}< 1$

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1994 p4

February 20, 2020
inequalitiesSequencealgebra

Problem Statement

Let a1,a2,...a_1,a_2,... be a sequence of natural numbers such that for each nn, the product (an−1)(an−2)...(an−n2)(a_n - 1)(a_n- 2)...(a_n - n^2) is a positive integral multiple of nn2−1n^{n^2-1}. Prove that for any finite set PP of prime numbers the following inequality holds: ∑p∈P1log⁡pap<1\sum_{p\in P}\frac{1}{\log_p a_p}< 1
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