Question 5
Two players $A$ and $B$ play a game in which they choose numbers alternately according to the following rule: At the beginning, an initial natural number $n_0\gt 1$ is given. Knowing $n_{2k}$, player $A$ may choose any $n_{2k+1}\in\mathbb N$ such that
$$n_{2k}\le n_{2k+1}\le n_{2k}^2.$$
Then player $B$ chooses a number $n_{2k+2}\in\mathbb N$ such that
$$\frac{n_{2k+1}}{n_{2k+2}}=p^r,$$
where $p$ is a prime number and $r\in\mathbb N$.
It is stipulated that player $A$ wins the game if he (she) succeeds in choosing the number 1990, and player $B$ wins if he (she) succeeds in choosing 1. For which natural numbers $n_0$ can player $A$ manage to win the game, for which $n_0$ can player $B$ manage to win, and for which $n_0$ can players $A$ and $B$ each force a tie?
$$n_{2k}\le n_{2k+1}\le n_{2k}^2.$$
Then player $B$ chooses a number $n_{2k+2}\in\mathbb N$ such that
$$\frac{n_{2k+1}}{n_{2k+2}}=p^r,$$
where $p$ is a prime number and $r\in\mathbb N$.
It is stipulated that player $A$ wins the game if he (she) succeeds in choosing the number 1990, and player $B$ wins if he (she) succeeds in choosing 1. For which natural numbers $n_0$ can player $A$ manage to win the game, for which $n_0$ can player $B$ manage to win, and for which $n_0$ can players $A$ and $B$ each force a tie?