Question 75
The expressions $x+y$, $x-y$, $x^2+xy+y^2$ and $x^2-xy+y^2$ are written on the two sides of two cards in such a way that each side of each card contains exactly one of these expressions. The cards are laid on the table on top of each other in such a way that only the top side of the uppermost card is visible. Alice and Bob, who know the expressions but not how they are distributed on the invisible sides of the cards, play the following game. Without inspecting the invisible sides of the cards, Alice picks one card according to her preference; the other card is left to Bob. Now both players may examine both sides of their card. Alice chooses a real value for either $x$ or $y$ according to her preference and tells her choice to Bob; then Bob chooses a real value for the other variable according to his preference. The player with the larger product of the values of the expressions on the two sides of their card wins. Does either of the players have a winning strategy, and if yes, then who does?