Problem

Source: Iran MO Third Round N1

Tags: polynomial, number theory, Rational roots



Assume natural number $n\ge2$. Amin and Ali take turns playing the following game: In each step, the player whose turn has come chooses index $i$ from the set $\{0,1,\cdots,n\}$, such that none of the two players had chosen this index in the previous turns; also this player in this turn chooses nonzero rational number $a_i$ too. Ali performs the first turn. The game ends when all the indices $i\in\{0,1,\cdots,n\}$ were chosen. In the end, from the chosen numbers the following polynomial is built: $$P(x)=a_nx^n+\cdots+a_1x+a_0$$Ali's goal is that the preceding polynomial has a rational root and Amin's goal is that to prevent this matter. Find all $n\ge2$ such that Ali can play in a way to be sure independent of how Amin plays achieves his goal.