International Mathematical Olympiad 1990 Problem 5

Given an initial integer $n_0 > 1$, two players, $\mathcal{A}$ and $\mathcal{B}$, choose integers $n_1, n_2, n_3, \ldots$ alternately according to the following rules:

Knowing $n_{2k}$, $\mathcal{A}$ chooses any integer $n_{2k+1}$ such that

$$n_{2k} \leq n_{2k+1} \leq n_{2k}^2.$$

Knowing $n_{2k+1}$, $\mathcal{B}$ chooses any integer $n_{2k+2}$ such that

$$\frac{n_{2k+1}}{n_{2k+2}}$$

is a prime raised to a positive integer power.

Player $\mathcal{A}$ wins the game by choosing the number 1990; player $\mathcal{B}$ wins by choosing the number 1. For which $n_0$ does:

(a) $\mathcal{A}$ have a winning strategy?

(b) $\mathcal{B}$ have a winning strategy?

(c) Neither player have a winning strategy?