CZ-SK-MO 2011 Problem 4

Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$ , $b\ge 2$ , $c\ge 2$ . Adam and Boris play the following game. They alternately take turns with Adam first. On Adam's turn, he can choose one of the polynomial's coefficients and replace it with the sum of the other two coefficients. On Boris's turn, he can choose one of the polynomial's coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$ , $b$ and $c$ , determine who has a winning strategy.