Question 64
Consider a natural number $n$. Amin and Ali play a game; each player at his own turn chooses a number $i$ from the set $\{0, 1, 2, \dots, n\}$ that has never been chosen before, and chooses a non-zero rational number $a_i$. Ali takes the first move. When all the numbers $\{0, 1, 2, \dots, n\}$ have been chosen, they form a polynomial $P(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_0$. Ali wins if the polynomial has a rational root, Amin wins otherwise. Find all $n$ such that Ali has a winning strategy.