4

Part of 1994 Czech And Slovak Olympiad IIIA

Problems(1)

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

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

a_1,a_2,...

be a sequence of natural numbers such that for each

n

(a_n - 1)(a_n- 2)...(a_n - n^2)

is a positive integral multiple of

n^{n^2-1}

. Prove that for any finite set

P

of prime numbers the following inequality holds:

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

inequalitiesSequencealgebra