Baltic Way 2019 · Problem 6

Alice and Bob play the following game. They write the expressions $x+y$, $x-y$, $x^2+xy+y^2$ and $x^2-xy+y^2$ each on a separate card. The four cards are shuffled and placed face down on a table. One of the cards is turned over, revealing the expression written on it, after which Alice chooses any two of the four cards, and gives the other two to Bob. All cards are then revealed. Now Alice picks one of the variables $x$ and $y$, assigns a real value to it, and tells Bob what value she assigned and to which variable. Then Bob assigns a real value to the other variable.

Finally, they both evaluate the product of the expressions on their two cards. Whoever gets the larger result wins. Which player, if any, has a winning strategy?