CZ-SK-MO 1994 Problem 4

Let $a_1,a_2,...$ be a sequence of natural numbers such that for each $n$ , the product $(a_n - 1)(a_n- 2)...(a_n - n^2)$ is a positive integral multiple of $n^{n^2-1}$ . Prove that for any finite set $P$ of prime numbers the following inequality holds: $$\sum_{p\in P}\frac{1}{\log_p a_p}< 1$$