Problem

Source: Iran MO Third Round N1

Tags: polynomial, number theory, Rational roots



Assume natural number n2. 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,,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 ai too. Ali performs the first turn. The game ends when all the indices i{0,1,,n} were chosen. In the end, from the chosen numbers the following polynomial is built: P(x)=anxn++a1x+a0Ali's goal is that the preceding polynomial has a rational root and Amin's goal is that to prevent this matter. Find all n2 such that Ali can play in a way to be sure independent of how Amin plays achieves his goal.