Question 8
Tim and Tam play a game. To start the game, Tim writes some nonzero real numbers, not necessarily distinct, on a blackboard. In each round, the following happens:
First, Tam chooses a polynomial $ P_{k}(x) = a_{k}x^{k} + a_{k-1}x^{k-1} + \cdots + a_{1}x + a_{0} $ whose coefficients $ a_{k}, a_{k-1}, \ldots, a_{1}, a_{0} $ are all of the numbers currently written on the blackboard in some order. (For example, if the numbers on the blackboard are 4, -3, 4, then k = 2 and Tam may choose the polynomial $ 4x^{2} - 3x + 4 $ or $ -3x^{2} + 4x + 4 $ but not $ 4x - 3 $ or $ -3x^{2} - 3x + 4 $.)
• Then, if the equation $P_{k}(x)=0$ has no real solutions, the game is stopped. Otherwise, Tim chooses a real number $r$ such that $P_{k}(r)=0$ and writes it on the blackboard, so there is now one more number on the blackboard.
Determine whether Tim can ensure that the game is never stopped, no matter what Tam does.
First, Tam chooses a polynomial $ P_{k}(x) = a_{k}x^{k} + a_{k-1}x^{k-1} + \cdots + a_{1}x + a_{0} $ whose coefficients $ a_{k}, a_{k-1}, \ldots, a_{1}, a_{0} $ are all of the numbers currently written on the blackboard in some order. (For example, if the numbers on the blackboard are 4, -3, 4, then k = 2 and Tam may choose the polynomial $ 4x^{2} - 3x + 4 $ or $ -3x^{2} + 4x + 4 $ but not $ 4x - 3 $ or $ -3x^{2} - 3x + 4 $.)
• Then, if the equation $P_{k}(x)=0$ has no real solutions, the game is stopped. Otherwise, Tim chooses a real number $r$ such that $P_{k}(r)=0$ and writes it on the blackboard, so there is now one more number on the blackboard.
Determine whether Tim can ensure that the game is never stopped, no matter what Tam does.