Question 3
Let $n\ge2$ be a positive integer. Consider the polynomial $1+\square x+\square x^2+\cdots+\square x^{2000}$, with 2000 empty boxes. Players A and B alternately enter a real number into an unfilled box, A first, until all boxes are filled. Write the resulting polynomial as $P(x)=1+a_1x+\cdots+a_{2000}x^{2000}$. The player making the last move loses if $P$ has a real root $\alpha$ with $|\alpha|\lt 1$, and wins otherwise. Prove that one of the two players has a strategy guaranteeing a win.
Editorial note: the initial $n$ is printed in the source but is not used; the game has exactly 2000 boxes.
Editorial note: the initial $n$ is printed in the source but is not used; the game has exactly 2000 boxes.