IMC 2012 Problem 1

Consider a polynomial $$f(x)=x^{2012}+a_{2011}x^{2011}+\dots+a_1x+a_0.$$ Albert Einstein and Homer Simpson are playing the following game. In turn, they choose one of the coefficients $a_0,a_1,\dots,a_{2011}$ and assign a real value to it. Albert has the first move. Once a value is assigned to a coefficient, it cannot be changed any more. The game ends after all the coefficients have been assigned values. Homer's goal is to make $f(x)$ divisible by a fixed polynomial $m(x)$ and Albert's goal is to prevent this. (a) Which of the players has a winning strategy if $m(x)=x-2012$ ? (b) Which of the players has a winning strategy if $m(x)=x^2+1$ ?

Proposed by Fedor Duzhin, Nanyang Technological University.