Question 134
A quadratic equation $x^2 + px + q = 0$ is written on the blackboard, whereby $p$ and $q$ are real numbers such that real solutions exist to the equation on the blackboard and all the solutions are positive. Two players change in turns the coefficients in the equation according to the following rules. The first player decreases the constant term by either solution of the equation and (on the same move) increases the coefficient at the linear term by 1. The second player may replace the constant term with an arbitrary real number. Alternatively, the second player may increase the constant term by the largest solution of the equation and (on the same move) decrease the coefficient at the linear term by 1, but such move is allowed only if the solutions of the equation on the blackboard before the move differ from each other by more than 1. If either player's move results in an equation that does not have real solutions or has a non-positive real solution then the first player wins. Can the first player win regardless of how the opponent plays?